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