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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-epelg | Structured version Visualization version GIF version |
Description: The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5576 and closed form of epeli 5575. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) TODO: move it to the main section after reordering to have brrelex1i 5725 available. (Proof shortened by BJ, 14-Jul-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-epelg | ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rele 5820 | . . . 4 ⊢ Rel E | |
2 | 1 | brrelex1i 5725 | . . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V) |
3 | 2 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V)) |
4 | elex 3487 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
5 | 4 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V)) |
6 | eleq12 2817 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵)) | |
7 | df-eprel 5573 | . . . 4 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} | |
8 | 6, 7 | brabga 5527 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
9 | 8 | expcom 413 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))) |
10 | 3, 5, 9 | pm5.21ndd 379 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2098 Vcvv 3468 class class class wbr 5141 E cep 5572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-eprel 5573 df-xp 5675 df-rel 5676 |
This theorem is referenced by: (None) |
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