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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-epelb | Structured version Visualization version GIF version |
Description: Two classes are related by the membership relation if and only if they are related by the membership relation (i.e., the first is an element of the second) and the second is a set (hence so is the first). TODO: move to Main after reordering to have brrelex2i 5733 available. Check if it is shorter to prove bj-epelg 36252 first or bj-epelb 36253 first. (Contributed by BJ, 14-Jul-2023.) |
Ref | Expression |
---|---|
bj-epelb | ⊢ (𝐴 E 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rele 5827 | . . . 4 ⊢ Rel E | |
2 | 1 | brrelex2i 5733 | . . 3 ⊢ (𝐴 E 𝐵 → 𝐵 ∈ V) |
3 | 2 | pm4.71i 560 | . 2 ⊢ (𝐴 E 𝐵 ↔ (𝐴 E 𝐵 ∧ 𝐵 ∈ V)) |
4 | epelg 5581 | . . 3 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
5 | 4 | pm5.32ri 576 | . 2 ⊢ ((𝐴 E 𝐵 ∧ 𝐵 ∈ V) ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V)) |
6 | 3, 5 | bitri 274 | 1 ⊢ (𝐴 E 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∈ wcel 2106 Vcvv 3474 class class class wbr 5148 E cep 5579 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-eprel 5580 df-xp 5682 df-rel 5683 |
This theorem is referenced by: (None) |
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