Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > opelopabb | Structured version Visualization version GIF version |
Description: Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.) |
Ref | Expression |
---|---|
opelopabb.xph | ⊢ (𝜑 → ∀𝑥𝜑) |
opelopabb.yph | ⊢ (𝜑 → ∀𝑦𝜑) |
opelopabb.xch | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
opelopabb.ych | ⊢ (𝜑 → Ⅎ𝑦𝜒) |
opelopabb.is | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
opelopabb | ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopab 5440 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜓)) | |
2 | opelopabb.xph | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | opelopabb.yph | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
4 | opelopabb.xch | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
5 | opelopabb.ych | . . 3 ⊢ (𝜑 → Ⅎ𝑦𝜒) | |
6 | opelopabb.is | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) | |
7 | 2, 3, 4, 5, 6 | copsex2b 35311 | . 2 ⊢ (𝜑 → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) |
8 | 1, 7 | syl5bb 283 | 1 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1537 = wceq 1539 ∃wex 1782 Ⅎwnf 1786 ∈ wcel 2106 Vcvv 3432 〈cop 4567 {copab 5136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-opab 5137 |
This theorem is referenced by: opelopabbv 35314 |
Copyright terms: Public domain | W3C validator |