Metamath Proof Explorer


Theorem prclaxpr

Description: A class that is closed under the pairing operation models the Axiom of Pairing ax-pr . Lemma II.2.4(4) of Kunen2 p. 111. (Contributed by Eric Schmidt, 29-Sep-2025)

Ref Expression
Assertion prclaxpr x M y M x y M x M y M z M w M w = x w = y w z

Proof

Step Hyp Ref Expression
1 vex w V
2 1 elpr w x y w = x w = y
3 2 biimpri w = x w = y w x y
4 3 rgenw w M w = x w = y w x y
5 eleq2 z = x y w z w x y
6 5 imbi2d z = x y w = x w = y w z w = x w = y w x y
7 6 ralbidv z = x y w M w = x w = y w z w M w = x w = y w x y
8 7 rspcev x y M w M w = x w = y w x y z M w M w = x w = y w z
9 4 8 mpan2 x y M z M w M w = x w = y w z
10 9 2ralimi x M y M x y M x M y M z M w M w = x w = y w z