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Theorem axpr 4213
 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 4214 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.)
Assertion
Ref Expression
axpr
Distinct variable groups:   ,,   ,,

Proof of Theorem axpr
StepHypRef Expression
1 zfpair 4212 . . 3
21isseti 2794 . 2
3 dfcleq 2277 . . . 4
4 vex 2791 . . . . . . . 8
54elpr 3658 . . . . . . 7
65bibi2i 304 . . . . . 6
7 bi2 189 . . . . . 6
86, 7sylbi 187 . . . . 5
98alimi 1546 . . . 4
103, 9sylbi 187 . . 3
1110eximi 1563 . 2
122, 11ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wo 357  wal 1527  wex 1528   wceq 1623   wcel 1684  cpr 3641 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647
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