Theorem xpsndisj 4339

Description: Cross products with two different singletons are disjoint.

Assertion

Ref	Expression
xpsndisj	|- (B =/= D -> ((A X. {B}) i^i (C X. {D})) = (/))

Proof of Theorem xpsndisj

Step	Hyp	Ref	Expression
1		disjsn2 3091	. 2 |- (B =/= D -> ({B} i^i {D}) = (/))
2		xpdisj2 4338	. 2 |- (({B} i^i {D}) = (/) -> ((A X. {B}) i^i (C X. {D})) = (/))
3	1, 2	syl 12	1 |- (B =/= D -> ((A X. {B}) i^i (C X. {D})) = (/))

Syntax hints:   -> wi 3   = wceq 1298   =/= wne 2017   i^i cin 2592  (/)c0 2875  {csn 3044   X. cxp 3984

This theorem is referenced by:  xp01disj 5188  unxpdom2 5997  sucxpdom 5998  cdacomen 6079

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-xp 4000  df-rel 4001  df-cnv 4002

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