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Theorem xp11 5056
 Description: The cross product of non-empty classes is one-to-one. (Contributed by set.mm contributors, 31-May-2008.)
Assertion
Ref Expression
xp11 ((A B) → ((A × B) = (C × D) ↔ (A = C B = D)))

Proof of Theorem xp11
StepHypRef Expression
1 xpnz 5045 . . 3 ((A B) ↔ (A × B) ≠ )
2 anidm 625 . . . . . 6 (((A × B) ≠ (A × B) ≠ ) ↔ (A × B) ≠ )
3 neeq1 2524 . . . . . . 7 ((A × B) = (C × D) → ((A × B) ≠ ↔ (C × D) ≠ ))
43anbi2d 684 . . . . . 6 ((A × B) = (C × D) → (((A × B) ≠ (A × B) ≠ ) ↔ ((A × B) ≠ (C × D) ≠ )))
52, 4syl5bbr 250 . . . . 5 ((A × B) = (C × D) → ((A × B) ≠ ↔ ((A × B) ≠ (C × D) ≠ )))
6 eqimss 3323 . . . . . . . 8 ((A × B) = (C × D) → (A × B) (C × D))
7 ssxpb 5055 . . . . . . . 8 ((A × B) ≠ → ((A × B) (C × D) ↔ (A C B D)))
86, 7syl5ibcom 211 . . . . . . 7 ((A × B) = (C × D) → ((A × B) ≠ → (A C B D)))
9 eqimss2 3324 . . . . . . . 8 ((A × B) = (C × D) → (C × D) (A × B))
10 ssxpb 5055 . . . . . . . 8 ((C × D) ≠ → ((C × D) (A × B) ↔ (C A D B)))
119, 10syl5ibcom 211 . . . . . . 7 ((A × B) = (C × D) → ((C × D) ≠ → (C A D B)))
128, 11anim12d 546 . . . . . 6 ((A × B) = (C × D) → (((A × B) ≠ (C × D) ≠ ) → ((A C B D) (C A D B))))
13 an4 797 . . . . . . 7 (((A C B D) (C A D B)) ↔ ((A C C A) (B D D B)))
14 eqss 3287 . . . . . . . 8 (A = C ↔ (A C C A))
15 eqss 3287 . . . . . . . 8 (B = D ↔ (B D D B))
1614, 15anbi12i 678 . . . . . . 7 ((A = C B = D) ↔ ((A C C A) (B D D B)))
1713, 16bitr4i 243 . . . . . 6 (((A C B D) (C A D B)) ↔ (A = C B = D))
1812, 17syl6ib 217 . . . . 5 ((A × B) = (C × D) → (((A × B) ≠ (C × D) ≠ ) → (A = C B = D)))
195, 18sylbid 206 . . . 4 ((A × B) = (C × D) → ((A × B) ≠ → (A = C B = D)))
2019com12 27 . . 3 ((A × B) ≠ → ((A × B) = (C × D) → (A = C B = D)))
211, 20sylbi 187 . 2 ((A B) → ((A × B) = (C × D) → (A = C B = D)))
22 xpeq12 4803 . 2 ((A = C B = D) → (A × B) = (C × D))
2321, 22impbid1 194 1 ((A B) → ((A × B) = (C × D) ↔ (A = C B = D)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ≠ wne 2516   ⊆ wss 3257  ∅c0 3550   × cxp 4770 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787 This theorem is referenced by:  xpcan  5057  xpcan2  5058
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