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Mirrors > Home > ILE Home > Th. List > elsn2g | GIF version |
Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that 𝐵, rather than 𝐴, be a set. (Contributed by NM, 28-Oct-2003.) |
Ref | Expression |
---|---|
elsn2g | ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsni 3416 | . 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | |
2 | snidg 3423 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵}) | |
3 | eleq1 2141 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝐵} ↔ 𝐵 ∈ {𝐵})) | |
4 | 2, 3 | syl5ibrcom 155 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ∈ {𝐵})) |
5 | 1, 4 | impbid2 141 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 {csn 3398 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sn 3404 |
This theorem is referenced by: elsn2 3428 elsuc2g 4160 mptiniseg 4835 elfzp1 9089 fzosplitsni 9244 iseqid3 9465 |
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