Theorem xp11 5057

Description: The cross product of nonempty 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

Step	Hyp	Ref	Expression
1		xpnz 5046	. . 3 ⊢ ((A ≠ ∅ ∧ B ≠ ∅) ↔ (A × B) ≠ ∅)
2		anidm 625	. . . . . 6 ⊢ (((A × B) ≠ ∅ ∧ (A × B) ≠ ∅) ↔ (A × B) ≠ ∅)
3		neeq1 2525	. . . . . . 7 ⊢ ((A × B) = (C × D) → ((A × B) ≠ ∅ ↔ (C × D) ≠ ∅))
4	3	anbi2d 684	. . . . . 6 ⊢ ((A × B) = (C × D) → (((A × B) ≠ ∅ ∧ (A × B) ≠ ∅) ↔ ((A × B) ≠ ∅ ∧ (C × D) ≠ ∅)))
5	2, 4	syl5bbr 250	. . . . 5 ⊢ ((A × B) = (C × D) → ((A × B) ≠ ∅ ↔ ((A × B) ≠ ∅ ∧ (C × D) ≠ ∅)))
6		eqimss 3324	. . . . . . . 8 ⊢ ((A × B) = (C × D) → (A × B) ⊆ (C × D))
7		ssxpb 5056	. . . . . . . 8 ⊢ ((A × B) ≠ ∅ → ((A × B) ⊆ (C × D) ↔ (A ⊆ C ∧ B ⊆ D)))
8	6, 7	syl5ibcom 211	. . . . . . 7 ⊢ ((A × B) = (C × D) → ((A × B) ≠ ∅ → (A ⊆ C ∧ B ⊆ D)))
9		eqimss2 3325	. . . . . . . 8 ⊢ ((A × B) = (C × D) → (C × D) ⊆ (A × B))
10		ssxpb 5056	. . . . . . . 8 ⊢ ((C × D) ≠ ∅ → ((C × D) ⊆ (A × B) ↔ (C ⊆ A ∧ D ⊆ B)))
11	9, 10	syl5ibcom 211	. . . . . . 7 ⊢ ((A × B) = (C × D) → ((C × D) ≠ ∅ → (C ⊆ A ∧ D ⊆ B)))
12	8, 11	anim12d 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 3288	. . . . . . . 8 ⊢ (A = C ↔ (A ⊆ C ∧ C ⊆ A))
15		eqss 3288	. . . . . . . 8 ⊢ (B = D ↔ (B ⊆ D ∧ D ⊆ B))
16	14, 15	anbi12i 678	. . . . . . 7 ⊢ ((A = C ∧ B = D) ↔ ((A ⊆ C ∧ C ⊆ A) ∧ (B ⊆ D ∧ D ⊆ B)))
17	13, 16	bitr4i 243	. . . . . 6 ⊢ (((A ⊆ C ∧ B ⊆ D) ∧ (C ⊆ A ∧ D ⊆ B)) ↔ (A = C ∧ B = D))
18	12, 17	syl6ib 217	. . . . 5 ⊢ ((A × B) = (C × D) → (((A × B) ≠ ∅ ∧ (C × D) ≠ ∅) → (A = C ∧ B = D)))
19	5, 18	sylbid 206	. . . 4 ⊢ ((A × B) = (C × D) → ((A × B) ≠ ∅ → (A = C ∧ B = D)))
20	19	com12 27	. . 3 ⊢ ((A × B) ≠ ∅ → ((A × B) = (C × D) → (A = C ∧ B = D)))
21	1, 20	sylbi 187	. 2 ⊢ ((A ≠ ∅ ∧ B ≠ ∅) → ((A × B) = (C × D) → (A = C ∧ B = D)))
22		xpeq12 4804	. 2 ⊢ ((A = C ∧ B = D) → (A × B) = (C × D))
23	21, 22	impbid1 194	1 ⊢ ((A ≠ ∅ ∧ B ≠ ∅) → ((A × B) = (C × D) ↔ (A = C ∧ B = D)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ≠ wne 2517   ⊆ wss 3258  ∅c0 3551   × cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788

This theorem is referenced by:  xpcan  5058  xpcan2  5059