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Theorem onxpdisj 5764
 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 5763. (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
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 disj 3969 . 2 ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V))
2 on0eqel 5762 . . 3 (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥))
3 0nelxp 5067 . . . . 5 ¬ ∅ ∈ (V × V)
4 eleq1 2676 . . . . 5 (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
53, 4mtbiri 316 . . . 4 (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V))
6 0nelelxp 5069 . . . . 5 (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥)
76con2i 133 . . . 4 (∅ ∈ 𝑥 → ¬ 𝑥 ∈ (V × V))
85, 7jaoi 393 . . 3 ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) → ¬ 𝑥 ∈ (V × V))
92, 8syl 17 . 2 (𝑥 ∈ On → ¬ 𝑥 ∈ (V × V))
101, 9mprgbir 2911 1 (On ∩ (V × V)) = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 382   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ∅c0 3874   × cxp 5036  Oncon0 5640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-ord 5643  df-on 5644 This theorem is referenced by: (None)
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