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Theorem zfpair2 4640
 Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 4639. See zfpair 4637 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)
Assertion
Ref Expression
zfpair2

Proof of Theorem zfpair2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-pr 4639 . . . 4
21bm1.3ii 4528 . . 3
3 dfcleq 2445 . . . . 5
4 vex 3048 . . . . . . . 8
54elpr 3986 . . . . . . 7
65bibi2i 315 . . . . . 6
76albii 1691 . . . . 5
83, 7bitri 253 . . . 4
98exbii 1718 . . 3
102, 9mpbir 213 . 2
1110issetri 3052 1
 Colors of variables: wff setvar class Syntax hints:   wb 188   wo 370  wal 1442   wceq 1444  wex 1663   wcel 1887  cvv 3045  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-sep 4525  ax-pr 4639 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  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-v 3047  df-un 3409  df-sn 3969  df-pr 3971 This theorem is referenced by:  snex  4641  prex  4642  pwssun  4740  xpsspw  4948  funopg  5614  fiint  7848  brdom7disj  8959  brdom6disj  8960  wlkntrllem1  25289  frisusgranb  25725  2pthfrgrarn  25737
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