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Theorem axpr 4638
 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 4639 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 4637 . . 3
21isseti 3051 . 2
3 dfcleq 2445 . . 3
4 vex 3048 . . . . . . 7
54elpr 3986 . . . . . 6
65bibi2i 315 . . . . 5
7 biimpr 202 . . . . 5
86, 7sylbi 199 . . . 4
98alimi 1684 . . 3
103, 9sylbi 199 . 2
112, 10eximii 1709 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 188   wo 370  wal 1442   wceq 1444  wex 1663   wcel 1887  cpr 3970 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-pw 3953  df-sn 3969  df-pr 3971 This theorem is referenced by: (None)
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