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Mirrors > Home > MPE Home > Th. List > axpr | Unicode version |
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 4691 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfpair 4689 | . . 3 | |
2 | 1 | isseti 3115 | . 2 |
3 | dfcleq 2450 | . . 3 | |
4 | vex 3112 | . . . . . . 7 | |
5 | 4 | elpr 4047 | . . . . . 6 |
6 | 5 | bibi2i 313 | . . . . 5 |
7 | bi2 198 | . . . . 5 | |
8 | 6, 7 | sylbi 195 | . . . 4 |
9 | 8 | alimi 1633 | . . 3 |
10 | 3, 9 | sylbi 195 | . 2 |
11 | 2, 10 | eximii 1658 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 <-> wb 184
\/ wo 368 A. wal 1393 = wceq 1395
E. wex 1612 e. wcel 1818 { cpr 4031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-rep 4563 ax-sep 4573 ax-nul 4581 ax-pow 4630 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-v 3111 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-pw 4014 df-sn 4030 df-pr 4032 |
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