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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 3550 | . 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | |
2 | snidg 3561 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵}) | |
3 | eleq1 2203 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ {𝐵} ↔ 𝐵 ∈ {𝐵})) | |
4 | 2, 3 | syl5ibrcom 156 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ∈ {𝐵})) |
5 | 1, 4 | impbid2 142 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ∈ wcel 1481 {csn 3532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-sn 3538 |
This theorem is referenced by: elsn2 3566 elsuc2g 4335 mptiniseg 5041 elfzp1 9883 fzosplitsni 10043 zfz1isolemiso 10614 |
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