![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5592 and closed form of epeli 5591. Definition 1.6 of [Schloeder] p. 1. (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 5149 | . . . 4 ⊢ (𝐴 E 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ E ) | |
2 | 0nelopab 5577 | . . . . . . . 8 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
3 | df-eprel 5589 | . . . . . . . . . 10 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
4 | 3 | eqcomi 2744 | . . . . . . . . 9 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} = E |
5 | 4 | eleq2i 2831 | . . . . . . . 8 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} ↔ ∅ ∈ E ) |
6 | 2, 5 | mtbi 322 | . . . . . . 7 ⊢ ¬ ∅ ∈ E |
7 | eleq1 2827 | . . . . . . 7 ⊢ (〈𝐴, 𝐵〉 = ∅ → (〈𝐴, 𝐵〉 ∈ E ↔ ∅ ∈ E )) | |
8 | 6, 7 | mtbiri 327 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 = ∅ → ¬ 〈𝐴, 𝐵〉 ∈ E ) |
9 | 8 | con2i 139 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ E → ¬ 〈𝐴, 𝐵〉 = ∅) |
10 | opprc1 4902 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → 〈𝐴, 𝐵〉 = ∅) | |
11 | 9, 10 | nsyl2 141 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ E → 𝐴 ∈ V) |
12 | 1, 11 | sylbi 217 | . . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V) |
13 | 12 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V)) |
14 | elex 3499 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
15 | 14 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V)) |
16 | eleq12 2829 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵)) | |
17 | 16, 3 | brabga 5544 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
18 | 17 | expcom 413 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))) |
19 | 13, 15, 18 | pm5.21ndd 379 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 〈cop 4637 class class class wbr 5148 {copab 5210 E cep 5588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-eprel 5589 |
This theorem is referenced by: epeli 5591 efrirr 5669 efrn2lp 5670 epin 6116 predep 6353 epne3 7792 cnfcomlem 9737 fpwwe2lem5 10673 ltpiord 10925 orvcelval 34450 bj-epelb 37052 brcnvep 38247 onsupuni 43218 oninfint 43225 onepsuc 43241 cantnfresb 43314 epelon2 43511 alephiso2 43548 |
Copyright terms: Public domain | W3C validator |