Theorem onxpdisj 3247

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 3131.

Assertion

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

Proof of Theorem onxpdisj

Step	Hyp	Ref	Expression
1		disj 2315	. 2 |- ((On i^i (V X. V)) = (/) <-> A.x e. On -. x e. (V X. V))
2		on0eqelt 3130	. . 3 |- (x e. On -> (x = (/) \/ (/) e. x))
3		0nelxp 3246	. . . . 5 |- -. (/) e. (V X. V)
4		eleq1 1537	. . . . 5 |- (x = (/) -> (x e. (V X. V) <-> (/) e. (V X. V)))
5	3, 4	mtbiri 719	. . . 4 |- (x = (/) -> -. x e. (V X. V))
6		elvv 3234	. . . . . 6 |- (x e. (V X. V) <-> E.yE.z x = <.y, z>.)
7		visset 1816	. . . . . . . . 9 |- y e. V
8		opprc1b 2802	. . . . . . . . . 10 |- (-. y e. V <-> (/) e. <.y, z>.)
9	8	con1bii 220	. . . . . . . . 9 |- (-. (/) e. <.y, z>. <-> y e. V)
10	7, 9	mpbir 190	. . . . . . . 8 |- -. (/) e. <.y, z>.
11		eleq2 1538	. . . . . . . 8 |- (x = <.y, z>. -> ((/) e. x <-> (/) e. <.y, z>.))
12	10, 11	mtbiri 719	. . . . . . 7 |- (x = <.y, z>. -> -. (/) e. x)
13	12	19.23aivv 1298	. . . . . 6 |- (E.yE.z x = <.y, z>. -> -. (/) e. x)
14	6, 13	sylbi 199	. . . . 5 |- (x e. (V X. V) -> -. (/) e. x)
15	14	con2i 97	. . . 4 |- ((/) e. x -> -. x e. (V X. V))
16	5, 15	jaoi 341	. . 3 |- ((x = (/) \/ (/) e. x) -> -. x e. (V X. V))
17	2, 16	syl 10	. 2 |- (x e. On -> -. x e. (V X. V))
18	1, 17	mprgbir 1704	1 |- (On i^i (V X. V)) = (/)

Syntax hints:  -. wn 2   \/ wo 222   = wceq 958   e. wcel 960  E.wex 982  Vcvv 1814   i^i cin 2049  (/)c0 2283  <.cop 2415  Oncon0 2954   X. cxp 3174

This theorem is referenced by:  onnev 3248

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-xp 3190

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