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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 4601 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴}) | |
2 | elex 3514 | . 2 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ V) | |
3 | 1, 2 | impbii 211 | 1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 Vcvv 3496 {csn 4569 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-sn 4570 |
This theorem is referenced by: snid 4603 dffv2 6758 snen1el 39898 |
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