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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 4663 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴}) | |
2 | elex 3490 | . 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 2099 Vcvv 3471 {csn 4629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 df-sn 4630 |
This theorem is referenced by: snid 4665 dffv2 6993 snen1el 42955 |
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