Theorem xpeq0 3473

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 3472	. . 3 |- ((A =/= (/) /\ B =/= (/)) <-> (A X. B) =/= (/))
2	1	necon2bbii 1624	. 2 |- ((A X. B) = (/) <-> -. (A =/= (/) /\ B =/= (/)))
3		ianor 305	. 2 |- (-. (A =/= (/) /\ B =/= (/)) <-> (-. A =/= (/) \/ -. B =/= (/)))
4		nne 1592	. . 3 |- (-. A =/= (/) <-> A = (/))
5		nne 1592	. . 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 958   =/= wne 1588  (/)c0 2283   X. cxp 3174

This theorem is referenced by:  rankxplim3 4724

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192

Copyright terms: Public domain