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Theorem onxpdisj 6009
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 6008. (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 4161 . 2 ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V))
2 on0eqel 6007 . . 3 (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥))
3 0nelxp 5301 . . . . 5 ¬ ∅ ∈ (V × V)
4 eleq1 2828 . . . . 5 (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
53, 4mtbiri 316 . . . 4 (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V))
6 0nelelxp 5303 . . . . 5 (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥)
76con2i 134 . . . 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 3066 1 (On ∩ (V × V)) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 382   = wceq 1632  wcel 2140  Vcvv 3341  cin 3715  c0 4059   × cxp 5265  Oncon0 5885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-tr 4906  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-ord 5888  df-on 5889
This theorem is referenced by: (None)
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