Theorem onxpdisj 4068

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 3788. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
onxpdisj	|- (On i^i (_V X. _V)) = (/)

Proof of Theorem onxpdisj

Step	Hyp	Ref	Expression
1		disj 2914	. 2 |- ((On i^i (_V X. _V)) = (/) <-> A.x e. On -. x e. (_V X. _V))
2		on0eqel 3787	. . 3 |- (x e. On -> (x = (/) \/ (/) e. x))
3		0nelxp 4066	. . . . 5 |- -. (/) e. (_V X. _V)
4		eleq1 1957	. . . . 5 |- (x = (/) -> (x e. (_V X. _V) <-> (/) e. (_V X. _V)))
5	3, 4	mtbiri 785	. . . 4 |- (x = (/) -> -. x e. (_V X. _V))
6		0nelelxp 4067	. . . . 5 |- (x e. (_V X. _V) -> -. (/) e. x)
7	6	con2i 113	. . . 4 |- ((/) e. x -> -. x e. (_V X. _V))
8	5, 7	jaoi 368	. . 3 |- ((x = (/) \/ (/) e. x) -> -. x e. (_V X. _V))
9	2, 8	syl 12	. 2 |- (x e. On -> -. x e. (_V X. _V))
10	1, 9	mprgbir 2163	1 |- (On i^i (_V X. _V)) = (/)

Syntax hints:  -. wn 2   \/ wo 239   = wceq 1298   e. wcel 1300  _Vcvv 2292   i^i cin 2592  (/)c0 2875  Oncon0 3657   X. cxp 3984

This theorem is referenced by:  onnev 4070

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-xp 4000

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