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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 5587 and closed form of epeli 5586. 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 5144 | . . . 4 ⊢ (𝐴 E 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ E ) | |
| 2 | 0nelopab 5572 | . . . . . . . 8 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
| 3 | df-eprel 5584 | . . . . . . . . . 10 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
| 4 | 3 | eqcomi 2746 | . . . . . . . . 9 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} = E |
| 5 | 4 | eleq2i 2833 | . . . . . . . 8 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} ↔ ∅ ∈ E ) |
| 6 | 2, 5 | mtbi 322 | . . . . . . 7 ⊢ ¬ ∅ ∈ E |
| 7 | eleq1 2829 | . . . . . . 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 4897 | . . . . 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 3501 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
| 15 | 14 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V)) |
| 16 | eleq12 2831 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵)) | |
| 17 | 16, 3 | brabga 5539 | . . 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 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 class class class wbr 5143 {copab 5205 E cep 5583 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 |
| This theorem is referenced by: epeli 5586 efrirr 5665 efrn2lp 5666 epin 6113 predep 6351 epne3 7793 cnfcomlem 9739 fpwwe2lem5 10675 ltpiord 10927 orvcelval 34471 bj-epelb 37070 brcnvep 38266 onsupuni 43241 oninfint 43248 onepsuc 43264 cantnfresb 43337 epelon2 43534 alephiso2 43571 |
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