Theorem xp11 3482

Description: The cross product of non-empty classes is one-to-one.

Assertion

Ref	Expression
xp11	|- ((A =/= (/) /\ B =/= (/)) -> ((A X. B) = (C X. D) <-> (A = C /\ B = D)))

Proof of Theorem xp11

Step	Hyp	Ref	Expression
1		xpnz 3472	. . 3 |- ((A =/= (/) /\ B =/= (/)) <-> (A X. B) =/= (/))
2		neeq1 1593	. . . . . . 7 |- ((A X. B) = (C X. D) -> ((A X. B) =/= (/) <-> (C X. D) =/= (/)))
3	2	anbi2d 618	. . . . . 6 |- ((A X. B) = (C X. D) -> (((A X. B) =/= (/) /\ (A X. B) =/= (/)) <-> ((A X. B) =/= (/) /\ (C X. D) =/= (/))))
4		anidm 434	. . . . . 6 |- (((A X. B) =/= (/) /\ (A X. B) =/= (/)) <-> (A X. B) =/= (/))
5	3, 4	syl5bbr 536	. . . . 5 |- ((A X. B) = (C X. D) -> ((A X. B) =/= (/) <-> ((A X. B) =/= (/) /\ (C X. D) =/= (/))))
6		eqimss 2112	. . . . . . . 8 |- ((A X. B) = (C X. D) -> (A X. B) (_ (C X. D))
7		ssxpr 3481	. . . . . . . . 9 |- (((A X. B) =/= (/) /\ (A X. B) (_ (C X. D)) -> (A (_ C /\ B (_ D))
8	7	expcom 374	. . . . . . . 8 |- ((A X. B) (_ (C X. D) -> ((A X. B) =/= (/) -> (A (_ C /\ B (_ D)))
9	6, 8	syl 10	. . . . . . 7 |- ((A X. B) = (C X. D) -> ((A X. B) =/= (/) -> (A (_ C /\ B (_ D)))
10		eqimss2 2113	. . . . . . . 8 |- ((A X. B) = (C X. D) -> (C X. D) (_ (A X. B))
11		ssxpr 3481	. . . . . . . . 9 |- (((C X. D) =/= (/) /\ (C X. D) (_ (A X. B)) -> (C (_ A /\ D (_ B))
12	11	expcom 374	. . . . . . . 8 |- ((C X. D) (_ (A X. B) -> ((C X. D) =/= (/) -> (C (_ A /\ D (_ B)))
13	10, 12	syl 10	. . . . . . 7 |- ((A X. B) = (C X. D) -> ((C X. D) =/= (/) -> (C (_ A /\ D (_ B)))
14	9, 13	anim12d 560	. . . . . 6 |- ((A X. B) = (C X. D) -> (((A X. B) =/= (/) /\ (C X. D) =/= (/)) -> ((A (_ C /\ B (_ D) /\ (C (_ A /\ D (_ B))))
15		an4 508	. . . . . . 7 |- (((A (_ C /\ B (_ D) /\ (C (_ A /\ D (_ B)) <-> ((A (_ C /\ C (_ A) /\ (B (_ D /\ D (_ B)))
16		eqss 2080	. . . . . . . 8 |- (A = C <-> (A (_ C /\ C (_ A))
17		eqss 2080	. . . . . . . 8 |- (B = D <-> (B (_ D /\ D (_ B))
18	16, 17	anbi12i 484	. . . . . . 7 |- ((A = C /\ B = D) <-> ((A (_ C /\ C (_ A) /\ (B (_ D /\ D (_ B)))
19	15, 18	bitr4 176	. . . . . 6 |- (((A (_ C /\ B (_ D) /\ (C (_ A /\ D (_ B)) <-> (A = C /\ B = D))
20	14, 19	syl6ib 212	. . . . 5 |- ((A X. B) = (C X. D) -> (((A X. B) =/= (/) /\ (C X. D) =/= (/)) -> (A = C /\ B = D)))
21	5, 20	sylbid 203	. . . 4 |- ((A X. B) = (C X. D) -> ((A X. B) =/= (/) -> (A = C /\ B = D)))
22	21	com12 11	. . 3 |- ((A X. B) =/= (/) -> ((A X. B) = (C X. D) -> (A = C /\ B = D)))
23	1, 22	sylbi 199	. 2 |- ((A =/= (/) /\ B =/= (/)) -> ((A X. B) = (C X. D) -> (A = C /\ B = D)))
24		xpeq1 3206	. . 3 |- (A = C -> (A X. B) = (C X. B))
25		xpeq2 3207	. . 3 |- (B = D -> (C X. B) = (C X. D))
26	24, 25	sylan9eq 1530	. 2 |- ((A = C /\ B = D) -> (A X. B) = (C X. D))
27	23, 26	impbid1 519	1 |- ((A =/= (/) /\ B =/= (/)) -> ((A X. B) = (C X. D) <-> (A = C /\ B = D)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   =/= wne 1588   (_ wss 2050  (/)c0 2283   X. cxp 3174

This theorem is referenced by:  xp11a 3483  xp11b 3484

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-9 967  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-ral 1652  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  df-dm 3194  df-rn 3195

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