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Theorem onxpdisj 5088
 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 5001. (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 3867 . 2
2 on0eqel 5000 . . 3
3 0nelxp 5032 . . . . 5
4 eleq1 2529 . . . . 5
53, 4mtbiri 303 . . . 4
6 0nelelxp 5033 . . . . 5
76con2i 120 . . . 4
85, 7jaoi 379 . . 3
92, 8syl 16 . 2
101, 9mprgbir 2821 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  \/wo 368  =wceq 1395  e.wcel 1818   cvv 3109  i^icin 3474   c0 3784   con0 4883  X.cxp 5002 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-xp 5010
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