Theorem xp11 3482

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

Assertion

Ref	Expression
xp11	⊢ ((A ≠ ∅ ⋀ B ≠ ∅) → ((A × B) = (C × D) ↔ (A = C ⋀ B = D)))

Proof of Theorem xp11

Step	Hyp	Ref	Expression
1		xpnz 3472	. . 3 ⊢ ((A ≠ ∅ ⋀ B ≠ ∅) ↔ (A × B) ≠ ∅)
2		neeq1 1593	. . . . . . 7 ⊢ ((A × B) = (C × D) → ((A × B) ≠ ∅ ↔ (C × D) ≠ ∅))
3	2	anbi2d 618	. . . . . 6 ⊢ ((A × B) = (C × D) → (((A × B) ≠ ∅ ⋀ (A × B) ≠ ∅) ↔ ((A × B) ≠ ∅ ⋀ (C × D) ≠ ∅)))
4		anidm 434	. . . . . 6 ⊢ (((A × B) ≠ ∅ ⋀ (A × B) ≠ ∅) ↔ (A × B) ≠ ∅)
5	3, 4	syl5bbr 536	. . . . 5 ⊢ ((A × B) = (C × D) → ((A × B) ≠ ∅ ↔ ((A × B) ≠ ∅ ⋀ (C × D) ≠ ∅)))
6		eqimss 2112	. . . . . . . 8 ⊢ ((A × B) = (C × D) → (A × B) ⊆ (C × D))
7		ssxpr 3481	. . . . . . . . 9 ⊢ (((A × B) ≠ ∅ ⋀ (A × B) ⊆ (C × D)) → (A ⊆ C ⋀ B ⊆ D))
8	7	expcom 374	. . . . . . . 8 ⊢ ((A × B) ⊆ (C × D) → ((A × B) ≠ ∅ → (A ⊆ C ⋀ B ⊆ D)))
9	6, 8	syl 10	. . . . . . 7 ⊢ ((A × B) = (C × D) → ((A × B) ≠ ∅ → (A ⊆ C ⋀ B ⊆ D)))
10		eqimss2 2113	. . . . . . . 8 ⊢ ((A × B) = (C × D) → (C × D) ⊆ (A × B))
11		ssxpr 3481	. . . . . . . . 9 ⊢ (((C × D) ≠ ∅ ⋀ (C × D) ⊆ (A × B)) → (C ⊆ A ⋀ D ⊆ B))
12	11	expcom 374	. . . . . . . 8 ⊢ ((C × D) ⊆ (A × B) → ((C × D) ≠ ∅ → (C ⊆ A ⋀ D ⊆ B)))
13	10, 12	syl 10	. . . . . . 7 ⊢ ((A × B) = (C × D) → ((C × D) ≠ ∅ → (C ⊆ A ⋀ D ⊆ B)))
14	9, 13	anim12d 560	. . . . . 6 ⊢ ((A × B) = (C × D) → (((A × B) ≠ ∅ ⋀ (C × 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 × B) = (C × D) → (((A × B) ≠ ∅ ⋀ (C × D) ≠ ∅) → (A = C ⋀ B = D)))
21	5, 20	sylbid 203	. . . 4 ⊢ ((A × B) = (C × D) → ((A × B) ≠ ∅ → (A = C ⋀ B = D)))
22	21	com12 11	. . 3 ⊢ ((A × B) ≠ ∅ → ((A × B) = (C × D) → (A = C ⋀ B = D)))
23	1, 22	sylbi 199	. 2 ⊢ ((A ≠ ∅ ⋀ B ≠ ∅) → ((A × B) = (C × D) → (A = C ⋀ B = D)))
24		xpeq1 3206	. . 3 ⊢ (A = C → (A × B) = (C × B))
25		xpeq2 3207	. . 3 ⊢ (B = D → (C × B) = (C × D))
26	24, 25	sylan9eq 1530	. 2 ⊢ ((A = C ⋀ B = D) → (A × B) = (C × D))
27	23, 26	impbid1 519	1 ⊢ ((A ≠ ∅ ⋀ B ≠ ∅) → ((A × B) = (C × D) ↔ (A = C ⋀ B = D)))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ≠ wne 1588   ⊆ wss 2050  ∅c0 2283   × 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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