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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elsnb | Structured version Visualization version GIF version |
Description: Biconditional version of elsng 4587. (Contributed by BJ, 18-Nov-2023.) |
Ref | Expression |
---|---|
bj-elsnb | ⊢ (𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3459 | . 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 ∈ V) | |
2 | elsng 4587 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | |
3 | 1, 2 | biadanii 819 | 1 ⊢ (𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 Vcvv 3441 {csn 4573 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3443 df-sn 4574 |
This theorem is referenced by: (None) |
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