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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 pairs. Compare to elopab 5379. (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 5379 | . 2 ⊢ (〈𝑢, 𝑣〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | vex 3444 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3444 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opth 5333 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝑢, 𝑣〉 ↔ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
5 | eqcom 2805 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝑢, 𝑣〉 ↔ 〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉) | |
6 | 4, 5 | bitr3i 280 | . . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ↔ 〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉) |
7 | 6 | anbi1i 626 | . . 3 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ (〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
8 | 7 | 2exbii 1850 | . 2 ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥∃𝑦(〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
9 | 1, 8 | bitr4i 281 | 1 ⊢ (〈𝑢, 𝑣〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 〈cop 4531 {copab 5092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 |
This theorem is referenced by: (None) |
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