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Breathe a sigh of relief over yesterday’s U.S. stock market rally from the brink of breakdown.

But then face reality. With all of the world’s major stock markets—except for the U.S. market–still sinking in search of some support, it’s extremely unlikely that U.S. stocks will be able to hold this ground alone.

Yesterday, May 25, the Standard & Poor’s 500 stock index bounced off of 1040 after an opening sell off. That level, 1040, marks the February low and acts as major support for the index. As long as the index stays above that level, investors can believe that this is no more than a correction. But if the index sinks below that February low, signaling that the market is now in search of a bottom at some lower level, then we’re in for another bout of selling on increased uncertainty and fear.

Yesterday the S&P 500, after threatening 1040 moved higher to close at 1074. That’s 34 points to the good.

Or maybe it’s better to say it’s just 34 points to the good. Because 34 points isn’t much margin for U.S. stocks when all the rest of the world’s major markets have broken below support.

Europe, of course, is in breakdown. As tracked by the iShares MSCI EAFE Index ETF (EFA), European stocks broke blow their February low back in April.

Asian shares have now followed suit and so in the last few days have the world’s emerging stock markets. The iShares MSCI Emerging Markets Index ETF (EEM) is now below its February low too.

Now this doesn’t guarantee that U.S. stocks will join the rout but you can see how much downward pressure the world’s other stock markets apply from a day like yesterday. The S&P 500 plunged at the beginning of trading under the momentum of a selloff in Asia that was followed by a selloff in Europe. By the time trading opened in New York, powerful momentum had built up to push U.S. stocks lower.

The example of yesterday, when good domestic news on consumer confidence and soothing words on financial reform legislation reversed a decline, shows that the S&P 500 index doesn’t have to succumb to this momentum. And U.S. stocks have moved down so far and so fast that a bargain-hunting bounce is likely. Such a bounce could take the index up just short of its April high of 1220 before failing.

But the odds are against the U.S. market being able to hold these levels alone. U.S. stocks need support from a turn in at least one of the two major negative stock market stories—the euro debt crisis or China growth slowdown fears—to stage a lasting comeback rather than a short-term bounce. Unfortunately, I don’t see a turn in either of those stories for months yet.

If the S&P 500 fails to hold at 1040, where’s the next likely stop? The next level, wrote technical analyst John Murphy in his Market Message on yesterday is 1010 on the S&P 500. That would be a 38% retracement of the gains in the rally. Next stop below that would be 945, he calculates. That’s a 50% retracement and also the May 2009 high.

But remember that just as rallies don’t go straight up hill without a pause, stock market drops are punctuated by small recoveries. I think we’re about due for one.

Of course, so does everyone else who can read a chart. That suggests that any rally is likely to be short because everyone will be looking to sell into it.