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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 5746 available. Check if it is shorter to prove bj-epelg 37051 first or bj-epelb 37052 first. (Contributed by BJ, 14-Jul-2023.) |
Ref | Expression |
---|---|
bj-epelb | ⊢ (𝐴 E 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rele 5840 | . . . 4 ⊢ Rel E | |
2 | 1 | brrelex2i 5746 | . . 3 ⊢ (𝐴 E 𝐵 → 𝐵 ∈ V) |
3 | 2 | pm4.71i 559 | . 2 ⊢ (𝐴 E 𝐵 ↔ (𝐴 E 𝐵 ∧ 𝐵 ∈ V)) |
4 | epelg 5590 | . . 3 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
5 | 4 | pm5.32ri 575 | . 2 ⊢ ((𝐴 E 𝐵 ∧ 𝐵 ∈ V) ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V)) |
6 | 3, 5 | bitri 275 | 1 ⊢ (𝐴 E 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 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-ral 3060 df-rex 3069 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 df-xp 5695 df-rel 5696 |
This theorem is referenced by: (None) |
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