![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > opelopabd | Structured version Visualization version GIF version |
Description: Membership of an ordere pair in a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.) |
Ref | Expression |
---|---|
opelopabd.xph | ⊢ (𝜑 → ∀𝑥𝜑) |
opelopabd.yph | ⊢ (𝜑 → ∀𝑦𝜑) |
opelopabd.xch | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
opelopabd.ych | ⊢ (𝜑 → Ⅎ𝑦𝜒) |
opelopabd.exa | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
opelopabd.exb | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
opelopabd.is | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
opelopabd | ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopab 5379 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜓)) | |
2 | opelopabd.xph | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | opelopabd.yph | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
4 | opelopabd.xch | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
5 | opelopabd.ych | . . 3 ⊢ (𝜑 → Ⅎ𝑦𝜒) | |
6 | opelopabd.exa | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
7 | opelopabd.exb | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
8 | opelopabd.is | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) | |
9 | 2, 3, 4, 5, 6, 7, 8 | copsex2d 34554 | . 2 ⊢ (𝜑 → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ 𝜒)) |
10 | 1, 9 | syl5bb 286 | 1 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 = wceq 1538 ∃wex 1781 Ⅎwnf 1785 ∈ 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-mo 2598 df-eu 2629 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: brabd0 34562 |
Copyright terms: Public domain | W3C validator |