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Mirrors > Home > MPE Home > Th. List > 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 5478 and closed form of epeli 5477. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 14-Jul-2023.) |
Ref | Expression |
---|---|
epelg | ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5069 | . . . 4 ⊢ (𝐴 E 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ E ) | |
2 | 0nelopab 5461 | . . . . . . . 8 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
3 | df-eprel 5475 | . . . . . . . . . 10 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
4 | 3 | eqcomi 2747 | . . . . . . . . 9 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} = E |
5 | 4 | eleq2i 2830 | . . . . . . . 8 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} ↔ ∅ ∈ E ) |
6 | 2, 5 | mtbi 325 | . . . . . . 7 ⊢ ¬ ∅ ∈ E |
7 | eleq1 2826 | . . . . . . 7 ⊢ (〈𝐴, 𝐵〉 = ∅ → (〈𝐴, 𝐵〉 ∈ E ↔ ∅ ∈ E )) | |
8 | 6, 7 | mtbiri 330 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 = ∅ → ¬ 〈𝐴, 𝐵〉 ∈ E ) |
9 | 8 | con2i 141 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ E → ¬ 〈𝐴, 𝐵〉 = ∅) |
10 | opprc1 4823 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → 〈𝐴, 𝐵〉 = ∅) | |
11 | 9, 10 | nsyl2 143 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ E → 𝐴 ∈ V) |
12 | 1, 11 | sylbi 220 | . . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V) |
13 | 12 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V)) |
14 | elex 3439 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
15 | 14 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V)) |
16 | eleq12 2828 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵)) | |
17 | 16, 3 | brabga 5430 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
18 | 17 | expcom 417 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))) |
19 | 13, 15, 18 | pm5.21ndd 384 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2111 Vcvv 3421 ∅c0 4252 〈cop 4562 class class class wbr 5068 {copab 5130 E cep 5474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ne 2942 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-br 5069 df-opab 5131 df-eprel 5475 |
This theorem is referenced by: epeli 5477 efrirr 5547 efrn2lp 5548 epin 5978 predep 6206 epne3 7577 cnfcomlem 9339 fpwwe2lem5 10274 ltpiord 10526 orvcelval 32174 bj-epelb 35005 brcnvep 36171 epelon2 40846 alephiso2 40874 |
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