Theorem onxpdisj 3236

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 3120.

Assertion

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

Proof of Theorem onxpdisj

Step	Hyp	Ref	Expression
1		disj 2307	. 2 |- ((On i^i (V X. V)) = (/) <-> A.x e. On -. x e. (V X. V))
2		on0eqelt 3119	. . 3 |- (x e. On -> (x = (/) \/ (/) e. x))
3		0nelxp 3235	. . . . 5 |- -. (/) e. (V X. V)
4		eleq1 1531	. . . . 5 |- (x = (/) -> (x e. (V X. V) <-> (/) e. (V X. V)))
5	3, 4	mtbiri 716	. . . 4 |- (x = (/) -> -. x e. (V X. V))
6		elvv 3223	. . . . . 6 |- (x e. (V X. V) <-> E.yE.z x = <.y, z>.)
7		visset 1809	. . . . . . . . 9 |- y e. V
8		opprc1b 2791	. . . . . . . . . 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 1532	. . . . . . . 8 |- (x = <.y, z>. -> ((/) e. x <-> (/) e. <.y, z>.))
12	10, 11	mtbiri 716	. . . . . . 7 |- (x = <.y, z>. -> -. (/) e. x)
13	12	19.23aivv 1294	. . . . . 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 1698	1 |- (On i^i (V X. V)) = (/)

Syntax hints:  -. wn 2   \/ wo 222   = wceq 954   e. wcel 956  E.wex 978  Vcvv 1807   i^i cin 2042  (/)c0 2276  <.cop 2407  Oncon0 2943   X. cxp 3163

This theorem is referenced by:  onnev 3237

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-xp 3179

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