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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 5469 and closed form of epeli 5468. (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 5608 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 5699 | . . . 4 ⊢ Rel E | |
2 | 1 | brrelex1i 5608 | . . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V) |
3 | 2 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V)) |
4 | elex 3512 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
5 | 4 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V)) |
6 | eleq12 2902 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵)) | |
7 | df-eprel 5465 | . . . 4 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
8 | 6, 7 | brabga 5421 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
9 | 8 | expcom 416 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))) |
10 | 3, 5, 9 | pm5.21ndd 383 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 E cep 5464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-eprel 5465 df-xp 5561 df-rel 5562 |
This theorem is referenced by: (None) |
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