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Mirrors > Home > ILE Home > Th. List > preqr2 | GIF version |
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
preqr2.1 | ⊢ 𝐴 ∈ V |
preqr2.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
preqr2 | ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 3657 | . . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶} | |
2 | prcom 3657 | . . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶} | |
3 | 1, 2 | eqeq12i 2184 | . 2 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶}) |
4 | preqr2.1 | . . 3 ⊢ 𝐴 ∈ V | |
5 | preqr2.2 | . . 3 ⊢ 𝐵 ∈ V | |
6 | 4, 5 | preqr1 3753 | . 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵) |
7 | 3, 6 | sylbi 120 | 1 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 Vcvv 2730 {cpr 3582 |
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-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 |
This theorem is referenced by: preq12b 3755 opth 4220 opthreg 4538 |
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