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Mirrors > Home > MPE Home > Th. List > elopabrOLD | Structured version Visualization version GIF version |
Description: Obsolete version of elopabr 5519 as of 11-Dec-2024. (Contributed by AV, 16-Feb-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elopabrOLD | ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopab 5485 | . 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)) | |
2 | df-br 5107 | . . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅) | |
3 | 2 | biimpi 215 | . . . . 5 ⊢ (𝑥𝑅𝑦 → ⟨𝑥, 𝑦⟩ ∈ 𝑅) |
4 | eleq1 2826 | . . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)) | |
5 | 3, 4 | imbitrrid 245 | . . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥𝑅𝑦 → 𝐴 ∈ 𝑅)) |
6 | 5 | imp 408 | . . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅) |
7 | 6 | exlimivv 1936 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅) |
8 | 1, 7 | sylbi 216 | 1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ⟨cop 4593 class class class wbr 5106 {copab 5168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 |
This theorem is referenced by: (None) |
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