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Mirrors > Home > MPE Home > Th. List > Mathboxes > opelopab4 | Structured version Visualization version GIF version |
Description: Ordered pair membership in a class abstraction of ordered pairs. Compare to elopab 5537. (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 5537 | . 2 ⊢ (〈𝑢, 𝑣〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | vex 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3482 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opth 5487 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝑢, 𝑣〉 ↔ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
5 | eqcom 2742 | . . . . 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 1846 | . 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 1537 ∃wex 1776 ∈ wcel 2106 〈cop 4637 {copab 5210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 |
This theorem is referenced by: (None) |
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