Theorem xp11b 3478

Description: The second argument of a cross product is one-to-one.

Assertion

Ref	Expression
xp11b	|- (A =/= (/) -> ((A X. A) = (A X. B) <-> A = B))

Proof of Theorem xp11b

Step	Hyp	Ref	Expression
1		xp11 3476	. . 3 |- ((A =/= (/) /\ A =/= (/)) -> ((A X. A) = (A X. B) <-> (A = A /\ A = B)))
2		eqid 1475	. . . 4 |- A = A
3	2	biantrur 725	. . 3 |- (A = B <-> (A = A /\ A = B))
4	1, 3	syl6bbr 538	. 2 |- ((A =/= (/) /\ A =/= (/)) -> ((A X. A) = (A X. B) <-> A = B))
5	4	anidms 434	1 |- (A =/= (/) -> ((A X. A) = (A X. B) <-> A = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   =/= wne 1585  (/)c0 2280   X. cxp 3168

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-9 965  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  df-dm 3188  df-rn 3189

Copyright terms: Public domain