Theorem xpid11 4834

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

Step	Hyp	Ref	Expression
1		dmeq 4811	. . 3 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) → dom (𝐴 × 𝐴) = dom (𝐵 × 𝐵))
2		dmxpid 4832	. . 3 ⊢ dom (𝐴 × 𝐴) = 𝐴
3		dmxpid 4832	. . 3 ⊢ dom (𝐵 × 𝐵) = 𝐵
4	1, 2, 3	3eqtr3g 2226	. 2 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) → 𝐴 = 𝐵)
5		xpeq12 4630	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) → (𝐴 × 𝐴) = (𝐵 × 𝐵))
6	5	anidms 395	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐴) = (𝐵 × 𝐵))
7	4, 6	impbii 125	1 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 104   = wceq 1348   × cxp 4609  dom cdm 4611

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-dm 4621

This theorem is referenced by:  intopsn  12621