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

Theorem onxpdisj 5835
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 5834. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
onxpdisj (On ∩ (V × V)) = ∅

Proof of Theorem onxpdisj
StepHypRef Expression
1 disj 4008 . 2 ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V))
2 on0eqel 5833 . . 3 (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥))
3 0nelxp 5133 . . . . 5 ¬ ∅ ∈ (V × V)
4 eleq1 2687 . . . . 5 (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
53, 4mtbiri 317 . . . 4 (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V))
6 0nelelxp 5135 . . . . 5 (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥)
76con2i 134 . . . 4 (∅ ∈ 𝑥 → ¬ 𝑥 ∈ (V × V))
85, 7jaoi 394 . . 3 ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) → ¬ 𝑥 ∈ (V × V))
92, 8syl 17 . 2 (𝑥 ∈ On → ¬ 𝑥 ∈ (V × V))
101, 9mprgbir 2924 1 (On ∩ (V × V)) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 383   = wceq 1481  wcel 1988  Vcvv 3195  cin 3566  c0 3907   × cxp 5102  Oncon0 5711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-ord 5714  df-on 5715
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator