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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 2897 | . 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 34456 | . 2 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) |
9 | 2, 8 | bitrd 281 | 1 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 〈cop 4566 {copab 5121 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5122 |
This theorem is referenced by: bj-opelidb 34466 |
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