Theorem xp01disj 5188

Description: Cross products with the singletons of ordinals 0 and 1 are disjoint.

Assertion

Ref	Expression
xp01disj	|- ((A X. {(/)}) i^i (C X. {1o})) = (/)

Proof of Theorem xp01disj

Step	Hyp	Ref	Expression
1		1n0 5187	. . 3 |- 1o =/= (/)
2		necom 2094	. . 3 |- (1o =/= (/) <-> (/) =/= 1o)
3	1, 2	mpbi 206	. 2 |- (/) =/= 1o
4		xpsndisj 4339	. 2 |- ((/) =/= 1o -> ((A X. {(/)}) i^i (C X. {1o})) = (/))
5	3, 4	ax-mp 7	1 |- ((A X. {(/)}) i^i (C X. {1o})) = (/)

Syntax hints:   = wceq 1298   =/= wne 2017   i^i cin 2592  (/)c0 2875  {csn 3044   X. cxp 3984  1oc1o 5172

This theorem is referenced by:  endisj 5496  uncdadom 6069  cdaun 6070  pm110.643 6072  cdaen 6073  cda1en 6076  cdacomen 6079  cdaassen 6080  xpcdaen 6081  mapcdaen 6082  cdadom1 6083

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-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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-1o 5177

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