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Mirrors > Home > ILE Home > Th. List > snidb | 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 3477 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴}) | |
2 | elex 2631 | . 2 ⊢ (𝐴 ∈ {𝐴} → 𝐴 ∈ V) | |
3 | 1, 2 | impbii 125 | 1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴}) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1439 Vcvv 2620 {csn 3450 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-sn 3456 |
This theorem is referenced by: snid 3479 |
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