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

Theorem onxpdisj 4960
 Description: Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 4703. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
onxpdisj

Proof of Theorem onxpdisj
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 disj 3670 . 2
2 on0eqel 4702 . . 3
3 0nelxp 4909 . . . . 5
4 eleq1 2498 . . . . 5
53, 4mtbiri 296 . . . 4
6 0nelelxp 4910 . . . . 5
76con2i 115 . . . 4
85, 7jaoi 370 . . 3
92, 8syl 16 . 2
101, 9mprgbir 2778 1
 Colors of variables: wff set class Syntax hints:   wn 3   wo 359   wceq 1653   wcel 1726  cvv 2958   cin 3321  c0 3630  con0 4584   cxp 4879 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-xp 4887
 Copyright terms: Public domain W3C validator