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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 4153 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴}) | |
2 | elex 3185 | . 2 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ V) | |
3 | 1, 2 | impbii 198 | 1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∈ wcel 1977 Vcvv 3173 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sn 4126 |
This theorem is referenced by: snid 4155 dffv2 6181 |
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