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Theorem axpr 4171
 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 4172 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 4170 . . 3
21isseti 2763 . 2
3 dfcleq 2250 . . . 4
4 vex 2760 . . . . . . . 8
54elpr 3618 . . . . . . 7
65bibi2i 306 . . . . . 6
7 bi2 191 . . . . . 6
86, 7sylbi 189 . . . . 5
98alimi 1546 . . . 4
103, 9sylbi 189 . . 3
1110eximi 1574 . 2
122, 11ax-mp 10 1
 Colors of variables: wff set class Syntax hints:   wi 6   wb 178   wo 359  wal 1532  wex 1537   wceq 1619   wcel 1621  cpr 3601 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-pw 3587  df-sn 3606  df-pr 3607
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