You can split the question into two parts: that the subsequent four cards will all be greater than 34, and that those four cards will be in ascending order.

The probability that the second card is greater than 34 is 66/99, then subsequent cards have a probability of 65/98, 64/97 and 63/96.

A lot of factors cancel, leaving an overall probability of 130/679 that the other four cards will be greater than the first.

Secondly, those four cards can be in any one of 4! Orders, only 1 of which is ascending, giving a factor of 1/24.

Combining these two probabilities we get an overall probability of 65/8148, which is about 1 in 125. Interestingly, this is slightly less than the probability if we were unaware of the number of the first card, which of course is 1 in 120.