Theorem xp11b 3475

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 3473	. . 3 |- ((A =/= (/) /\ A =/= (/)) -> ((A X. A) = (A X. B) <-> (A = A /\ A = B)))
2		eqid 1475	. . . 4 |- A = A
3	2	biantrur 724	. . 3 |- (A = B <-> (A = A /\ A = B))
4	1, 3	syl6bbr 537	. 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 955   =/= wne 1584  (/)c0 2278   X. cxp 3165

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664  df-xp 3181  df-rel 3182  df-cnv 3183  df-dm 3185  df-rn 3186

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