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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 5414 | . 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 34435 | . 2 ⊢ (𝜑 → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) |
8 | 1, 7 | syl5bb 285 | 1 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∃wex 1780 Ⅎwnf 1784 ∈ wcel 2114 Vcvv 3494 〈cop 4573 {copab 5128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 |
This theorem is referenced by: opelopabbv 34438 |
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