![]() |
Mathbox for Alan Sare |
< Previous
Next >
Nearby theorems |
|
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 5523. (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 5523 | . 2 ⊢ (〈𝑢, 𝑣〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | vex 3473 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3473 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opth 5472 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝑢, 𝑣〉 ↔ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
5 | eqcom 2734 | . . . . 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 623 | . . 3 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ↔ (〈𝑢, 𝑣〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
8 | 7 | 2exbii 1844 | . 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 205 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 〈cop 4630 {copab 5204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-opab 5205 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |