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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 3686 | . . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶} | |
2 | prcom 3686 | . . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶} | |
3 | 1, 2 | eqeq12i 2203 | . 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 3786 | . 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵) |
7 | 3, 6 | sylbi 121 | 1 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 Vcvv 2752 {cpr 3611 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-sn 3616 df-pr 3617 |
This theorem is referenced by: preq12b 3788 opth 4258 opthreg 4576 |
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