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Mirrors > Home > ILE Home > Th. List > elsn | GIF version |
Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) |
Ref | Expression |
---|---|
elsn.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elsn | ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | elsng 3390 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1243 ∈ wcel 1393 Vcvv 2557 {csn 3375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-sn 3381 |
This theorem is referenced by: velsn 3392 sneqr 3531 onsucelsucexmid 4255 ordsoexmid 4286 opthprc 4391 dmsnm 4786 dmsnopg 4792 cnvcnvsn 4797 sniota 4894 fsn 5335 eusvobj2 5498 opelreal 6904 |
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