Theorem xp11m 4686

Description: The cross product of inhabited classes is one-to-one. (Contributed by Jim Kingdon, 13-Dec-2018.)

Assertion

Ref	Expression
xp11m	⊢ ((∃x x ∈ A ∧ ∃y y ∈ B) → ((A × B) = (𝐶 × 𝐷) ↔ (A = 𝐶 ∧ B = 𝐷)))

Distinct variable groups:   x,A   y,B

Allowed substitution hints:   A(y)   B(x)   𝐶(x,y)   𝐷(x,y)

Proof of Theorem xp11m

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpm 4672	. . 3 ⊢ ((∃x x ∈ A ∧ ∃y y ∈ B) ↔ ∃z z ∈ (A × B))
2		anidm 376	. . . . . 6 ⊢ ((∃z z ∈ (A × B) ∧ ∃z z ∈ (A × B)) ↔ ∃z z ∈ (A × B))
3		eleq2 2083	. . . . . . . 8 ⊢ ((A × B) = (𝐶 × 𝐷) → (z ∈ (A × B) ↔ z ∈ (𝐶 × 𝐷)))
4	3	exbidv 1688	. . . . . . 7 ⊢ ((A × B) = (𝐶 × 𝐷) → (∃z z ∈ (A × B) ↔ ∃z z ∈ (𝐶 × 𝐷)))
5	4	anbi2d 440	. . . . . 6 ⊢ ((A × B) = (𝐶 × 𝐷) → ((∃z z ∈ (A × B) ∧ ∃z z ∈ (A × B)) ↔ (∃z z ∈ (A × B) ∧ ∃z z ∈ (𝐶 × 𝐷))))
6	2, 5	syl5bbr 183	. . . . 5 ⊢ ((A × B) = (𝐶 × 𝐷) → (∃z z ∈ (A × B) ↔ (∃z z ∈ (A × B) ∧ ∃z z ∈ (𝐶 × 𝐷))))
7		eqimss 2974	. . . . . . . 8 ⊢ ((A × B) = (𝐶 × 𝐷) → (A × B) ⊆ (𝐶 × 𝐷))
8		ssxpbm 4683	. . . . . . . 8 ⊢ (∃z z ∈ (A × B) → ((A × B) ⊆ (𝐶 × 𝐷) ↔ (A ⊆ 𝐶 ∧ B ⊆ 𝐷)))
9	7, 8	syl5ibcom 144	. . . . . . 7 ⊢ ((A × B) = (𝐶 × 𝐷) → (∃z z ∈ (A × B) → (A ⊆ 𝐶 ∧ B ⊆ 𝐷)))
10		eqimss2 2975	. . . . . . . 8 ⊢ ((A × B) = (𝐶 × 𝐷) → (𝐶 × 𝐷) ⊆ (A × B))
11		ssxpbm 4683	. . . . . . . 8 ⊢ (∃z z ∈ (𝐶 × 𝐷) → ((𝐶 × 𝐷) ⊆ (A × B) ↔ (𝐶 ⊆ A ∧ 𝐷 ⊆ B)))
12	10, 11	syl5ibcom 144	. . . . . . 7 ⊢ ((A × B) = (𝐶 × 𝐷) → (∃z z ∈ (𝐶 × 𝐷) → (𝐶 ⊆ A ∧ 𝐷 ⊆ B)))
13	9, 12	anim12d 318	. . . . . 6 ⊢ ((A × B) = (𝐶 × 𝐷) → ((∃z z ∈ (A × B) ∧ ∃z z ∈ (𝐶 × 𝐷)) → ((A ⊆ 𝐶 ∧ B ⊆ 𝐷) ∧ (𝐶 ⊆ A ∧ 𝐷 ⊆ B))))
14		an4 507	. . . . . . 7 ⊢ (((A ⊆ 𝐶 ∧ B ⊆ 𝐷) ∧ (𝐶 ⊆ A ∧ 𝐷 ⊆ B)) ↔ ((A ⊆ 𝐶 ∧ 𝐶 ⊆ A) ∧ (B ⊆ 𝐷 ∧ 𝐷 ⊆ B)))
15		eqss 2937	. . . . . . . 8 ⊢ (A = 𝐶 ↔ (A ⊆ 𝐶 ∧ 𝐶 ⊆ A))
16		eqss 2937	. . . . . . . 8 ⊢ (B = 𝐷 ↔ (B ⊆ 𝐷 ∧ 𝐷 ⊆ B))
17	15, 16	anbi12i 436	. . . . . . 7 ⊢ ((A = 𝐶 ∧ B = 𝐷) ↔ ((A ⊆ 𝐶 ∧ 𝐶 ⊆ A) ∧ (B ⊆ 𝐷 ∧ 𝐷 ⊆ B)))
18	14, 17	bitr4i 176	. . . . . 6 ⊢ (((A ⊆ 𝐶 ∧ B ⊆ 𝐷) ∧ (𝐶 ⊆ A ∧ 𝐷 ⊆ B)) ↔ (A = 𝐶 ∧ B = 𝐷))
19	13, 18	syl6ib 150	. . . . 5 ⊢ ((A × B) = (𝐶 × 𝐷) → ((∃z z ∈ (A × B) ∧ ∃z z ∈ (𝐶 × 𝐷)) → (A = 𝐶 ∧ B = 𝐷)))
20	6, 19	sylbid 139	. . . 4 ⊢ ((A × B) = (𝐶 × 𝐷) → (∃z z ∈ (A × B) → (A = 𝐶 ∧ B = 𝐷)))
21	20	com12 27	. . 3 ⊢ (∃z z ∈ (A × B) → ((A × B) = (𝐶 × 𝐷) → (A = 𝐶 ∧ B = 𝐷)))
22	1, 21	sylbi 114	. 2 ⊢ ((∃x x ∈ A ∧ ∃y y ∈ B) → ((A × B) = (𝐶 × 𝐷) → (A = 𝐶 ∧ B = 𝐷)))
23		xpeq12 4291	. 2 ⊢ ((A = 𝐶 ∧ B = 𝐷) → (A × B) = (𝐶 × 𝐷))
24	22, 23	impbid1 130	1 ⊢ ((∃x x ∈ A ∧ ∃y y ∈ B) → ((A × B) = (𝐶 × 𝐷) ↔ (A = 𝐶 ∧ B = 𝐷)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374   ⊆ wss 2894   × cxp 4270

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-dm 4282  df-rn 4283

This theorem is referenced by: (None)