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| Mirrors > Home > MPE Home > Th. List > Mathboxes > opelopabbv | 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 | 
|---|---|
| opelopabbv.def | ⊢ (𝜑 → 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓}) | 
| opelopabbv.is | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| opelopabbv | ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | opelopabbv.def | . . 3 ⊢ (𝜑 → 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓}) | |
| 2 | 1 | eleq2d 2827 | . 2 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓})) | 
| 3 | ax-5 1910 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
| 4 | ax-5 1910 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
| 5 | nfvd 1915 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
| 6 | nfvd 1915 | . . 3 ⊢ (𝜑 → Ⅎ𝑦𝜒) | |
| 7 | opelopabbv.is | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) | |
| 8 | 3, 4, 5, 6, 7 | opelopabb 37143 | . 2 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) | 
| 9 | 2, 8 | bitrd 279 | 1 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 {copab 5205 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 | 
| This theorem is referenced by: bj-opelidb 37153 | 
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