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Theorem onxpdisj 6512
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 6511. (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 4456 . 2 ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V))
2 on0eqel 6510 . . 3 (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥))
3 0nelxp 5723 . . . . 5 ¬ ∅ ∈ (V × V)
4 eleq1 2827 . . . . 5 (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
53, 4mtbiri 327 . . . 4 (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V))
6 0nelelxp 5724 . . . . 5 (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥)
76con2i 139 . . . 4 (∅ ∈ 𝑥 → ¬ 𝑥 ∈ (V × V))
85, 7jaoi 857 . . 3 ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) → ¬ 𝑥 ∈ (V × V))
92, 8syl 17 . 2 (𝑥 ∈ On → ¬ 𝑥 ∈ (V × V))
101, 9mprgbir 3066 1 (On ∩ (V × V)) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 847   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  c0 4339   × cxp 5687  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-ord 6389  df-on 6390
This theorem is referenced by: (None)
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