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Theorem onxpdisj 6061
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 6060. (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 4213 . 2 ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V))
2 on0eqel 6059 . . 3 (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥))
3 0nelxp 5347 . . . . 5 ¬ ∅ ∈ (V × V)
4 eleq1 2867 . . . . 5 (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
53, 4mtbiri 319 . . . 4 (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V))
6 0nelelxp 5348 . . . . 5 (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥)
76con2i 137 . . . 4 (∅ ∈ 𝑥 → ¬ 𝑥 ∈ (V × V))
85, 7jaoi 884 . . 3 ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) → ¬ 𝑥 ∈ (V × V))
92, 8syl 17 . 2 (𝑥 ∈ On → ¬ 𝑥 ∈ (V × V))
101, 9mprgbir 3109 1 (On ∩ (V × V)) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 874   = wceq 1653  wcel 2157  Vcvv 3386  cin 3769  c0 4116   × cxp 5311  Oncon0 5942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-tr 4947  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-xp 5319  df-ord 5945  df-on 5946
This theorem is referenced by: (None)
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