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Mirrors > Home > MPE Home > Th. List > Mathboxes > oppr | Structured version Visualization version GIF version |
Description: Equality for ordered pairs implies equality of unordered pairs with the same elements. (Contributed by AV, 9-Jul-2023.) |
Ref | Expression |
---|---|
oppr | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → {𝐴, 𝐵} = {𝐶, 𝐷})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opthg 5376 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
2 | preq12 4666 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) | |
3 | 1, 2 | syl6bi 256 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → {𝐴, 𝐵} = {𝐶, 𝐷})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 {cpr 4558 〈cop 4562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 |
This theorem is referenced by: or2expropbi 44231 ich2exprop 44627 2exopprim 44681 reuopreuprim 44682 |
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