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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 4908. (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 4908 | . 2 ⊢ (〈𝑢, 𝑣〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | vex 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opth 4871 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝑢, 𝑣〉 ↔ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
5 | eqcom 2617 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝑢, 𝑣〉 ↔ 〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉) | |
6 | 4, 5 | bitr3i 265 | . . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ↔ 〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉) |
7 | 6 | anbi1i 727 | . . 3 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ (〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
8 | 7 | 2exbii 1765 | . 2 ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥∃𝑦(〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
9 | 1, 8 | bitr4i 266 | 1 ⊢ (〈𝑢, 𝑣〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 〈cop 4131 {copab 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 |
This theorem is referenced by: (None) |
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