Theorem xpid11 5003

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 4979	. . 3 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) → dom (𝐴 × 𝐴) = dom (𝐵 × 𝐵))
2		dmxpid 5001	. . 3 ⊢ dom (𝐴 × 𝐴) = 𝐴
3		dmxpid 5001	. . 3 ⊢ dom (𝐵 × 𝐵) = 𝐵
4	1, 2, 3	3eqtr3g 2294	. 2 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) → 𝐴 = 𝐵)
5		xpeq12 4791	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) → (𝐴 × 𝐴) = (𝐵 × 𝐵))
6	5	anidms 401	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐴) = (𝐵 × 𝐵))
7	4, 6	impbii 126	1 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 105   = wceq 1402   × cxp 4770  dom cdm 4772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-dm 4782

This theorem is referenced by:  intopsn  13670