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Mirrors > Home > MPE Home > Th. List > elopabr | Structured version Visualization version GIF version |
Description: Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.) |
Ref | Expression |
---|---|
elopabr | ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopab 5382 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦)) | |
2 | df-br 5031 | . . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
3 | 2 | biimpi 219 | . . . . 5 ⊢ (𝑥𝑅𝑦 → 〈𝑥, 𝑦〉 ∈ 𝑅) |
4 | eleq1 2820 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝐴 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | |
5 | 3, 4 | syl5ibr 249 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝑥𝑅𝑦 → 𝐴 ∈ 𝑅)) |
6 | 5 | imp 410 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅) |
7 | 6 | exlimivv 1939 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅) |
8 | 1, 7 | sylbi 220 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2114 〈cop 4522 class class class wbr 5030 {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-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-sb 2075 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-br 5031 df-opab 5093 |
This theorem is referenced by: elopabran 5416 |
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