Theorem onxpdisj 2476

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 2371.

Assertion

Ref	Expression
onxpdisj	⊢ (On ∩ (V × V)) = ∅

Proof of Theorem onxpdisj

Step	Hyp	Ref	Expression
1		disj 1733	. 2 ⊢ ((On ∩ (V × V)) = ∅ ↔ ∀x ∈ On ¬ x ∈ (V × V))
2		on0eqelt 2370	. . 3 ⊢ (x ∈ On → (x = ∅ ∨ ∅ ∈ x))
3		0nelxp 2475	. . . . 5 ⊢ ¬ ∅ ∈ (V × V)
4		eleq1 1149	. . . . 5 ⊢ (x = ∅ → (x ∈ (V × V) ↔ ∅ ∈ (V × V)))
5	3, 4	mtbiri 539	. . . 4 ⊢ (x = ∅ → ¬ x ∈ (V × V))
6		elvv 2464	. . . . . 6 ⊢ (x ∈ (V × V) ↔ ∃y∃z x = ⟨y, z⟩)
7		visset 1350	. . . . . . . . 9 ⊢ y ∈ V
8		opprc1b 1906	. . . . . . . . . 10 ⊢ (¬ y ∈ V ↔ ∅ ∈ ⟨y, z⟩)
9	8	bicon1i 193	. . . . . . . . 9 ⊢ (¬ ∅ ∈ ⟨y, z⟩ ↔ y ∈ V)
10	7, 9	mpbir 165	. . . . . . . 8 ⊢ ¬ ∅ ∈ ⟨y, z⟩
11		eleq2 1150	. . . . . . . 8 ⊢ (x = ⟨y, z⟩ → (∅ ∈ x ↔ ∅ ∈ ⟨y, z⟩))
12	10, 11	mtbiri 539	. . . . . . 7 ⊢ (x = ⟨y, z⟩ → ¬ ∅ ∈ x)
13	12	19.23aivv 953	. . . . . 6 ⊢ (∃y∃z x = ⟨y, z⟩ → ¬ ∅ ∈ x)
14	6, 13	sylbi 174	. . . . 5 ⊢ (x ∈ (V × V) → ¬ ∅ ∈ x)
15	14	con2i 89	. . . 4 ⊢ (∅ ∈ x → ¬ x ∈ (V × V))
16	5, 15	jaoi 275	. . 3 ⊢ ((x = ∅ ∨ ∅ ∈ x) → ¬ x ∈ (V × V))
17	2, 16	syl 12	. 2 ⊢ (x ∈ On → ¬ x ∈ (V × V))
18	1, 17	mprgbir 1250	1 ⊢ (On ∩ (V × V)) = ∅

Syntax hints:  ¬ wn 1   ∨ wo 195  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∩ cin 1486  ∅c0 1707  ⟨cop 1810  Oncon0 2199   × cxp 2408

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-xp 2424

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