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Mirrors > Home > MPE Home > Th. List > preqr2 | Structured version Visualization version 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, 15-Jul-1993.) |
Ref | Expression |
---|---|
preqr1.a | ⊢ 𝐴 ∈ V |
preqr1.b | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
preqr2 | ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 4211 | . . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶} | |
2 | prcom 4211 | . . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶} | |
3 | 1, 2 | eqeq12i 2624 | . 2 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶}) |
4 | preqr1.a | . . 3 ⊢ 𝐴 ∈ V | |
5 | preqr1.b | . . 3 ⊢ 𝐵 ∈ V | |
6 | 4, 5 | preqr1 4319 | . 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵) |
7 | 3, 6 | sylbi 206 | 1 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-sn 4126 df-pr 4128 |
This theorem is referenced by: preq12b 4322 opth 4871 opthreg 8398 usgra2edg 25911 usgraedgreu 25917 nbgraf1olem5 25974 altopthsn 31238 usgredgreu 40445 uspgredg2vtxeu 40447 |
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