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| Description: Ordered pair membership in a class abstraction of ordered pairs. Compare to elopab 5531. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| opelopab4 | ⊢ (〈𝑢, 𝑣〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elopab 5531 | . 2 ⊢ (〈𝑢, 𝑣〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
| 2 | vex 3483 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 3 | vex 3483 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 4 | 2, 3 | opth 5480 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝑢, 𝑣〉 ↔ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) | 
| 5 | eqcom 2743 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝑢, 𝑣〉 ↔ 〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉) | |
| 6 | 4, 5 | bitr3i 277 | . . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ↔ 〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉) | 
| 7 | 6 | anbi1i 624 | . . 3 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ (〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | 
| 8 | 7 | 2exbii 1848 | . 2 ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥∃𝑦(〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | 
| 9 | 1, 8 | bitr4i 278 | 1 ⊢ (〈𝑢, 𝑣〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 〈cop 4631 {copab 5204 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 | 
| This theorem is referenced by: (None) | 
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