Theorem xpsndisj 3470

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 2442	. 2 |- (B =/= D -> ({B} i^i {D}) = (/))
2		xpdisj2 3469	. 2 |- (({B} i^i {D}) = (/) -> ((A X. {B}) i^i (C X. {D})) = (/))
3	1, 2	syl 10	1 |- (B =/= D -> ((A X. {B}) i^i (C X. {D})) = (/))

Syntax hints:   -> wi 3   = wceq 956   =/= wne 1585   i^i cin 2046  (/)c0 2280  {csn 2409   X. cxp 3168

This theorem is referenced by:  xp01disj 4143  unxpdom2 4845  sucxpdom 4846  cdacomen 4929

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186

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