Theorem xp11bOLD 4349

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

Assertion

Ref	Expression
xp11bOLD	|- (A =/= (/) -> ((A X. A) = (A X. B) <-> A = B))

Proof of Theorem xp11bOLD

Step	Hyp	Ref	Expression
1		xp11 4347	. . 3 |- ((A =/= (/) /\ A =/= (/)) -> ((A X. A) = (A X. B) <-> (A = A /\ A = B)))
2		eqid 1884	. . . 4 |- A = A
3	2	biantrur 794	. . 3 |- (A = B <-> (A = A /\ A = B))
4	1, 3	syl6bbr 597	. 2 |- ((A =/= (/) /\ A =/= (/)) -> ((A X. A) = (A X. B) <-> A = B))
5	4	anidms 480	1 |- (A =/= (/) -> ((A X. A) = (A X. B) <-> A = B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   =/= wne 2017  (/)c0 2875   X. cxp 3984

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005

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