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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 5433 and closed form of epeli 5432. (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 5031 | . . . 4 ⊢ (𝐴 E 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ E ) | |
2 | 0nelopab 5417 | . . . . . . . 8 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
3 | df-eprel 5430 | . . . . . . . . . 10 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
4 | 3 | eqcomi 2807 | . . . . . . . . 9 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} = E |
5 | 4 | eleq2i 2881 | . . . . . . . 8 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} ↔ ∅ ∈ E ) |
6 | 2, 5 | mtbi 325 | . . . . . . 7 ⊢ ¬ ∅ ∈ E |
7 | eleq1 2877 | . . . . . . 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 4789 | . . . . 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 3459 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
15 | 14 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V)) |
16 | eleq12 2879 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵)) | |
17 | 16, 3 | brabga 5386 | . . 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 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 〈cop 4531 class class class wbr 5030 {copab 5092 E cep 5429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-eprel 5430 |
This theorem is referenced by: epeli 5432 efrirr 5500 efrn2lp 5501 predep 6142 epne3 7475 cnfcomlem 9146 fpwwe2lem6 10046 ltpiord 10298 orvcelval 31836 bj-epelb 34485 brcnvep 35686 epelon2 40229 alephiso2 40257 |
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