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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 5382 | . 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 34952 | . 2 ⊢ (𝜑 → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) |
8 | 1, 7 | syl5bb 286 | 1 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1540 = wceq 1542 ∃wex 1786 Ⅎwnf 1790 ∈ wcel 2114 Vcvv 3398 〈cop 4522 {copab 5092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3400 df-dif 3846 df-un 3848 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-opab 5093 |
This theorem is referenced by: opelopabbv 34955 |
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