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Theorem xp11m 4702
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.
StepHypRef Expression
1 xpm 4688 . . 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 2098 . . . . . . . 8 ((A × B) = (𝐶 × 𝐷) → (z (A × B) ↔ z (𝐶 × 𝐷)))
43exbidv 1703 . . . . . . 7 ((A × B) = (𝐶 × 𝐷) → (z z (A × B) ↔ z z (𝐶 × 𝐷)))
54anbi2d 437 . . . . . 6 ((A × B) = (𝐶 × 𝐷) → ((z z (A × B) z z (A × B)) ↔ (z z (A × B) z z (𝐶 × 𝐷))))
62, 5syl5bbr 183 . . . . 5 ((A × B) = (𝐶 × 𝐷) → (z z (A × B) ↔ (z z (A × B) z z (𝐶 × 𝐷))))
7 eqimss 2991 . . . . . . . 8 ((A × B) = (𝐶 × 𝐷) → (A × B) ⊆ (𝐶 × 𝐷))
8 ssxpbm 4699 . . . . . . . 8 (z z (A × B) → ((A × B) ⊆ (𝐶 × 𝐷) ↔ (A𝐶 B𝐷)))
97, 8syl5ibcom 144 . . . . . . 7 ((A × B) = (𝐶 × 𝐷) → (z z (A × B) → (A𝐶 B𝐷)))
10 eqimss2 2992 . . . . . . . 8 ((A × B) = (𝐶 × 𝐷) → (𝐶 × 𝐷) ⊆ (A × B))
11 ssxpbm 4699 . . . . . . . 8 (z z (𝐶 × 𝐷) → ((𝐶 × 𝐷) ⊆ (A × B) ↔ (𝐶A 𝐷B)))
1210, 11syl5ibcom 144 . . . . . . 7 ((A × B) = (𝐶 × 𝐷) → (z z (𝐶 × 𝐷) → (𝐶A 𝐷B)))
139, 12anim12d 318 . . . . . 6 ((A × B) = (𝐶 × 𝐷) → ((z z (A × B) z z (𝐶 × 𝐷)) → ((A𝐶 B𝐷) (𝐶A 𝐷B))))
14 an4 520 . . . . . . 7 (((A𝐶 B𝐷) (𝐶A 𝐷B)) ↔ ((A𝐶 𝐶A) (B𝐷 𝐷B)))
15 eqss 2954 . . . . . . . 8 (A = 𝐶 ↔ (A𝐶 𝐶A))
16 eqss 2954 . . . . . . . 8 (B = 𝐷 ↔ (B𝐷 𝐷B))
1715, 16anbi12i 433 . . . . . . 7 ((A = 𝐶 B = 𝐷) ↔ ((A𝐶 𝐶A) (B𝐷 𝐷B)))
1814, 17bitr4i 176 . . . . . 6 (((A𝐶 B𝐷) (𝐶A 𝐷B)) ↔ (A = 𝐶 B = 𝐷))
1913, 18syl6ib 150 . . . . 5 ((A × B) = (𝐶 × 𝐷) → ((z z (A × B) z z (𝐶 × 𝐷)) → (A = 𝐶 B = 𝐷)))
206, 19sylbid 139 . . . 4 ((A × B) = (𝐶 × 𝐷) → (z z (A × B) → (A = 𝐶 B = 𝐷)))
2120com12 27 . . 3 (z z (A × B) → ((A × B) = (𝐶 × 𝐷) → (A = 𝐶 B = 𝐷)))
221, 21sylbi 114 . 2 ((x x A y y B) → ((A × B) = (𝐶 × 𝐷) → (A = 𝐶 B = 𝐷)))
23 xpeq12 4307 . 2 ((A = 𝐶 B = 𝐷) → (A × B) = (𝐶 × 𝐷))
2422, 23impbid1 130 1 ((x x A y y B) → ((A × B) = (𝐶 × 𝐷) ↔ (A = 𝐶 B = 𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  wss 2911   × cxp 4286
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by: (None)
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