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Theorem xpid11m 4500
Description: The cross product of a class with itself is one-to-one. (Contributed by Jim Kingdon, 8-Dec-2018.)
Assertion
Ref Expression
xpid11m ((x x A x x B) → ((A × A) = (B × B) ↔ A = B))
Distinct variable groups:   x,A   x,B

Proof of Theorem xpid11m
StepHypRef Expression
1 dmxpm 4498 . . . . . 6 (x x A → dom (A × A) = A)
21adantr 261 . . . . 5 ((x x A x x B) → dom (A × A) = A)
3 dmeq 4478 . . . . 5 ((A × A) = (B × B) → dom (A × A) = dom (B × B))
42, 3sylan9req 2090 . . . 4 (((x x A x x B) (A × A) = (B × B)) → A = dom (B × B))
5 dmxpm 4498 . . . . 5 (x x B → dom (B × B) = B)
65ad2antlr 458 . . . 4 (((x x A x x B) (A × A) = (B × B)) → dom (B × B) = B)
74, 6eqtrd 2069 . . 3 (((x x A x x B) (A × A) = (B × B)) → A = B)
87ex 108 . 2 ((x x A x x B) → ((A × A) = (B × B) → A = B))
9 xpeq12 4307 . . 3 ((A = B A = B) → (A × A) = (B × B))
109anidms 377 . 2 (A = B → (A × A) = (B × B))
118, 10impbid1 130 1 ((x x A x x B) → ((A × A) = (B × B) ↔ A = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390   × cxp 4286  dom cdm 4288
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-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-dm 4298
This theorem is referenced by: (None)
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