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Mirrors > Home > MPE Home > Th. List > snidb | Structured version Visualization version GIF version |
Description: A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.) |
Ref | Expression |
---|---|
snidb | ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snidg 4655 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴}) | |
2 | elex 3485 | . 2 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ V) | |
3 | 1, 2 | impbii 208 | 1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 Vcvv 3466 {csn 4621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-sn 4622 |
This theorem is referenced by: snid 4657 dffv2 6977 snen1el 42825 |
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