Theorem xpeq0 3467

Description: At least one member of an empty cross product is empty.

Assertion

Ref	Expression
xpeq0	|- ((A X. B) = (/) <-> (A = (/) \/ B = (/)))

Proof of Theorem xpeq0

Step	Hyp	Ref	Expression
1		xpnz 3466	. . 3 |- ((A =/= (/) /\ B =/= (/)) <-> (A X. B) =/= (/))
2	1	necon2bbii 1621	. 2 |- ((A X. B) = (/) <-> -. (A =/= (/) /\ B =/= (/)))
3		ianor 305	. 2 |- (-. (A =/= (/) /\ B =/= (/)) <-> (-. A =/= (/) \/ -. B =/= (/)))
4		nne 1589	. . 3 |- (-. A =/= (/) <-> A = (/))
5		nne 1589	. . 3 |- (-. B =/= (/) <-> B = (/))
6	4, 5	orbi12i 257	. 2 |- ((-. A =/= (/) \/ -. B =/= (/)) <-> (A = (/) \/ B = (/)))
7	2, 3, 6	3bitr 177	1 |- ((A X. B) = (/) <-> (A = (/) \/ B = (/)))

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956   =/= wne 1585  (/)c0 2280   X. cxp 3168

This theorem is referenced by:  rankxplim3 4714

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-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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