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Metamath Proof Explorer

Theorem prssad

Description: If a pair is a subset of a class, the first element of the pair is an element of that class. (Contributed by Thierry Arnoux, 2-Nov-2025)

Ref Expression
Hypotheses prssad.1 
|- ( ph -> A e. V )
prssad.2 
|- ( ph -> { A , B } C_ C )
Assertion prssad 
|- ( ph -> A e. C )

Proof

Step Hyp Ref Expression
1 prssad.1 
 |-  ( ph -> A e. V )
2 prssad.2 
 |-  ( ph -> { A , B } C_ C )
3 1 adantr 
 |-  ( ( ph /\ B e. _V ) -> A e. V )
4 simpr 
 |-  ( ( ph /\ B e. _V ) -> B e. _V )
5 2 adantr 
 |-  ( ( ph /\ B e. _V ) -> { A , B } C_ C )
6 prssg 
 |-  ( ( A e. V /\ B e. _V ) -> ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) )
7 6 biimpar 
 |-  ( ( ( A e. V /\ B e. _V ) /\ { A , B } C_ C ) -> ( A e. C /\ B e. C ) )
8 3 4 5 7 syl21anc 
 |-  ( ( ph /\ B e. _V ) -> ( A e. C /\ B e. C ) )
9 8 simpld 
 |-  ( ( ph /\ B e. _V ) -> A e. C )
10 prprc2 
 |-  ( -. B e. _V -> { A , B } = { A } )
11 10 adantl 
 |-  ( ( ph /\ -. B e. _V ) -> { A , B } = { A } )
12 2 adantr 
 |-  ( ( ph /\ -. B e. _V ) -> { A , B } C_ C )
13 11 12 eqsstrrd 
 |-  ( ( ph /\ -. B e. _V ) -> { A } C_ C )
14 snssg 
 |-  ( A e. V -> ( A e. C <-> { A } C_ C ) )
15 14 biimpar 
 |-  ( ( A e. V /\ { A } C_ C ) -> A e. C )
16 1 13 15 syl2an2r 
 |-  ( ( ph /\ -. B e. _V ) -> A e. C )
17 9 16 pm2.61dan 
 |-  ( ph -> A e. C )