Metamath Proof Explorer


Theorem axpr

Description: Unabbreviated version of the Axiom of Pairing of ZF set theory, derived as a theorem from the other axioms.

This theorem should not be referenced by any proof. Instead, use ax-pr below so that the uses of the Axiom of Pairing can be more easily identified.

For a shorter proof using ax-ext , see axprALT . (Contributed by NM, 14-Nov-2006) Remove dependency on ax-ext . (Revised by Rohan Ridenour, 10-Aug-2023) (Proof shortened by BJ, 13-Aug-2023) (Proof shortened by Matthew House, 18-Sep-2025) Use ax-pr instead. (New usage is discouraged.)

Ref Expression
Assertion axpr z w w = x w = y w z

Proof

Step Hyp Ref Expression
1 axprlem3 z w w z s s p if- n n s w = x w = y
2 axprlem1 s n t ¬ t n n s
3 2 sepexi s n n s t ¬ t n
4 biimp n s t ¬ t n n s t ¬ t n
5 ax-nul n t ¬ t n
6 exbi n n s t ¬ t n n n s n t ¬ t n
7 5 6 mpbiri n n s t ¬ t n n n s
8 ifptru n n s if- n n s w = x w = y w = x
9 7 8 syl n n s t ¬ t n if- n n s w = x w = y w = x
10 3 4 9 axprlem4 s n s t ¬ t n s p w = x s s p if- n n s w = x w = y
11 ax-nul s n ¬ n s
12 pm2.21 ¬ n s n s t ¬ t n
13 alnex n ¬ n s ¬ n n s
14 ifpfal ¬ n n s if- n n s w = x w = y w = y
15 13 14 sylbi n ¬ n s if- n n s w = x w = y w = y
16 11 12 15 axprlem4 s n s t ¬ t n s p w = y s s p if- n n s w = x w = y
17 10 16 jaod s n s t ¬ t n s p w = x w = y s s p if- n n s w = x w = y
18 imbi2 w z s s p if- n n s w = x w = y w = x w = y w z w = x w = y s s p if- n n s w = x w = y
19 17 18 syl5ibrcom s n s t ¬ t n s p w z s s p if- n n s w = x w = y w = x w = y w z
20 19 alimdv s n s t ¬ t n s p w w z s s p if- n n s w = x w = y w w = x w = y w z
21 20 eximdv s n s t ¬ t n s p z w w z s s p if- n n s w = x w = y z w w = x w = y w z
22 1 21 mpi s n s t ¬ t n s p z w w = x w = y w z
23 axprlem2 p s n s t ¬ t n s p
24 22 23 exlimiiv z w w = x w = y w z