Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  axpr Structured version   Unicode version

Theorem axpr 4402
 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 4403 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 4401 . . 3
21isseti 2962 . 2
3 dfcleq 2430 . . 3
4 vex 2959 . . . . . . 7
54elpr 3832 . . . . . 6
65bibi2i 305 . . . . 5
7 bi2 190 . . . . 5
86, 7sylbi 188 . . . 4
98alimi 1568 . . 3
103, 9sylbi 188 . 2
112, 10eximii 1587 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wo 358  wal 1549  wex 1550   wceq 1652   wcel 1725  cpr 3815 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-pw 3801  df-sn 3820  df-pr 3821
 Copyright terms: Public domain W3C validator