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Mirrors > Home > ILE Home > Th. List > xpid11 | GIF version |
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.) |
Ref | Expression |
---|---|
xpid11 | ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 4798 | . . 3 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) → dom (𝐴 × 𝐴) = dom (𝐵 × 𝐵)) | |
2 | dmxpid 4819 | . . 3 ⊢ dom (𝐴 × 𝐴) = 𝐴 | |
3 | dmxpid 4819 | . . 3 ⊢ dom (𝐵 × 𝐵) = 𝐵 | |
4 | 1, 2, 3 | 3eqtr3g 2220 | . 2 ⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) → 𝐴 = 𝐵) |
5 | xpeq12 4617 | . . 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) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1342 × cxp 4596 dom cdm 4598 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-dm 4608 |
This theorem is referenced by: (None) |
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