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Theorem xpid11 4774
 Description: The Cartesian product of a class with itself is one-to-one. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
xpid11 ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵)

Proof of Theorem xpid11
StepHypRef Expression
1 dmeq 4751 . . 3 ((𝐴 × 𝐴) = (𝐵 × 𝐵) → dom (𝐴 × 𝐴) = dom (𝐵 × 𝐵))
2 dmxpid 4772 . . 3 dom (𝐴 × 𝐴) = 𝐴
3 dmxpid 4772 . . 3 dom (𝐵 × 𝐵) = 𝐵
41, 2, 33eqtr3g 2197 . 2 ((𝐴 × 𝐴) = (𝐵 × 𝐵) → 𝐴 = 𝐵)
5 xpeq12 4570 . . 3 ((𝐴 = 𝐵𝐴 = 𝐵) → (𝐴 × 𝐴) = (𝐵 × 𝐵))
65anidms 395 . 2 (𝐴 = 𝐵 → (𝐴 × 𝐴) = (𝐵 × 𝐵))
74, 6impbii 125 1 ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1332   × cxp 4549  dom cdm 4551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1732  df-eu 1993  df-mo 1994  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-br 3940  df-opab 4000  df-xp 4557  df-dm 4561 This theorem is referenced by: (None)
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