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Theorem xpid11m 4654
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 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) → ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem xpid11m
StepHypRef Expression
1 dmxpm 4652 . . . . . 6 (∃𝑥 𝑥𝐴 → dom (𝐴 × 𝐴) = 𝐴)
21adantr 270 . . . . 5 ((∃𝑥 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) → dom (𝐴 × 𝐴) = 𝐴)
3 dmeq 4632 . . . . 5 ((𝐴 × 𝐴) = (𝐵 × 𝐵) → dom (𝐴 × 𝐴) = dom (𝐵 × 𝐵))
42, 3sylan9req 2141 . . . 4 (((∃𝑥 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) ∧ (𝐴 × 𝐴) = (𝐵 × 𝐵)) → 𝐴 = dom (𝐵 × 𝐵))
5 dmxpm 4652 . . . . 5 (∃𝑥 𝑥𝐵 → dom (𝐵 × 𝐵) = 𝐵)
65ad2antlr 473 . . . 4 (((∃𝑥 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) ∧ (𝐴 × 𝐴) = (𝐵 × 𝐵)) → dom (𝐵 × 𝐵) = 𝐵)
74, 6eqtrd 2120 . . 3 (((∃𝑥 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) ∧ (𝐴 × 𝐴) = (𝐵 × 𝐵)) → 𝐴 = 𝐵)
87ex 113 . 2 ((∃𝑥 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) → ((𝐴 × 𝐴) = (𝐵 × 𝐵) → 𝐴 = 𝐵))
9 xpeq12 4455 . . 3 ((𝐴 = 𝐵𝐴 = 𝐵) → (𝐴 × 𝐴) = (𝐵 × 𝐵))
109anidms 389 . 2 (𝐴 = 𝐵 → (𝐴 × 𝐴) = (𝐵 × 𝐵))
118, 10impbid1 140 1 ((∃𝑥 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) → ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438   × cxp 4434  dom cdm 4436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-br 3844  df-opab 3898  df-xp 4442  df-dm 4446
This theorem is referenced by: (None)
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