R1 Acceptees: Are you still applying to R2 schools?

I have a question. I will be grateful for the help.

I would like to mention to a school that I am applying for Round 2 (applying to school Y), the admit that I have received from a school I applied to in round 1 (admitted school X). The reason I want to do that is to try to leverage the school X admit. However my concern is that I do not want school Y to contact school X as it will make me look bad in front of school X for leveraging their admit in this way. If school Y needs to confirm my admittance in school X I can provide them other forms of irrefutable evidence: like signed admit letter from school X, also I have received my email address for school X so an email from my official school X email address which is only possible if I have an admit, etc

So my question is will school Y contact my admitted round 1 schools or will they be satisfied enough with the evidence I provide them of my admittance to school X (if they want it) ? I wonder if a school will also release to a competitor school whom it gave acceptance in round 1.

I know applicants use the admits from other schools to try to get financial aid and that is acceptable. I do not know whether one is allowed to do that with the admit itself. I think I can do this and am worrying too much but I just want to confirm.