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Mirrors > Home > MPE Home > Th. List > Mathboxes > opprb | Structured version Visualization version GIF version |
Description: Equality for unordered pairs corresponds to equality of unordered pairs with the same elements. (Contributed by AV, 9-Jul-2023.) |
Ref | Expression |
---|---|
opprb | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ∨ 〈𝐴, 𝐵〉 = 〈𝐷, 𝐶〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq12bg 4777 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))) | |
2 | opthg 5362 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
3 | 2 | adantr 483 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
4 | opthg 5362 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐷, 𝐶〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))) | |
5 | 4 | adantr 483 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (〈𝐴, 𝐵〉 = 〈𝐷, 𝐶〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))) |
6 | 3, 5 | orbi12d 915 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ((〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ∨ 〈𝐴, 𝐵〉 = 〈𝐷, 𝐶〉) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))) |
7 | 1, 6 | bitr4d 284 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ∨ 〈𝐴, 𝐵〉 = 〈𝐷, 𝐶〉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 {cpr 4562 〈cop 4566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 |
This theorem is referenced by: (None) |
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