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Theorem xpid11 5502
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 5479 . . 3 ((𝐴 × 𝐴) = (𝐵 × 𝐵) → dom (𝐴 × 𝐴) = dom (𝐵 × 𝐵))
2 dmxpid 5500 . . 3 dom (𝐴 × 𝐴) = 𝐴
3 dmxpid 5500 . . 3 dom (𝐵 × 𝐵) = 𝐵
41, 2, 33eqtr3g 2817 . 2 ((𝐴 × 𝐴) = (𝐵 × 𝐵) → 𝐴 = 𝐵)
5 xpeq12 5291 . . 3 ((𝐴 = 𝐵𝐴 = 𝐵) → (𝐴 × 𝐴) = (𝐵 × 𝐵))
65anidms 680 . 2 (𝐴 = 𝐵 → (𝐴 × 𝐴) = (𝐵 × 𝐵))
74, 6impbii 199 1 ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632   × cxp 5264  dom cdm 5266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-dm 5276
This theorem is referenced by:  intopsn  17454  grporn  27684  ismndo2  33986  rngosn3  34036  rngomndo  34047
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