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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.
StepHypRef 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)))
43exbidv 1688 . . . . . . 7 ((A × B) = (𝐶 × 𝐷) → (z z (A × B) ↔ z z (𝐶 × 𝐷)))
54anbi2d 440 . . . . . 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 2974 . . . . . . . 8 ((A × B) = (𝐶 × 𝐷) → (A × B) ⊆ (𝐶 × 𝐷))
8 ssxpbm 4683 . . . . . . . 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 2975 . . . . . . . 8 ((A × B) = (𝐶 × 𝐷) → (𝐶 × 𝐷) ⊆ (A × B))
11 ssxpbm 4683 . . . . . . . 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 507 . . . . . . 7 (((A𝐶 B𝐷) (𝐶A 𝐷B)) ↔ ((A𝐶 𝐶A) (B𝐷 𝐷B)))
15 eqss 2937 . . . . . . . 8 (A = 𝐶 ↔ (A𝐶 𝐶A))
16 eqss 2937 . . . . . . . 8 (B = 𝐷 ↔ (B𝐷 𝐷B))
1715, 16anbi12i 436 . . . . . . 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 4291 . 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 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)
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