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Theorem xpid11m 4585
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 4583 . . . . . 6 (∃𝑥 𝑥𝐴 → dom (𝐴 × 𝐴) = 𝐴)
21adantr 265 . . . . 5 ((∃𝑥 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) → dom (𝐴 × 𝐴) = 𝐴)
3 dmeq 4563 . . . . 5 ((𝐴 × 𝐴) = (𝐵 × 𝐵) → dom (𝐴 × 𝐴) = dom (𝐵 × 𝐵))
42, 3sylan9req 2109 . . . 4 (((∃𝑥 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) ∧ (𝐴 × 𝐴) = (𝐵 × 𝐵)) → 𝐴 = dom (𝐵 × 𝐵))
5 dmxpm 4583 . . . . 5 (∃𝑥 𝑥𝐵 → dom (𝐵 × 𝐵) = 𝐵)
65ad2antlr 466 . . . 4 (((∃𝑥 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) ∧ (𝐴 × 𝐴) = (𝐵 × 𝐵)) → dom (𝐵 × 𝐵) = 𝐵)
74, 6eqtrd 2088 . . 3 (((∃𝑥 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) ∧ (𝐴 × 𝐴) = (𝐵 × 𝐵)) → 𝐴 = 𝐵)
87ex 112 . 2 ((∃𝑥 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) → ((𝐴 × 𝐴) = (𝐵 × 𝐵) → 𝐴 = 𝐵))
9 xpeq12 4392 . . 3 ((𝐴 = 𝐵𝐴 = 𝐵) → (𝐴 × 𝐴) = (𝐵 × 𝐵))
109anidms 383 . 2 (𝐴 = 𝐵 → (𝐴 × 𝐴) = (𝐵 × 𝐵))
118, 10impbid1 134 1 ((∃𝑥 𝑥𝐴 ∧ ∃𝑥 𝑥𝐵) → ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409   × cxp 4371  dom cdm 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-dm 4383
This theorem is referenced by: (None)
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