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Mirrors > Home > ILE Home > Th. List > elsnc | GIF version |
Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) |
Ref | Expression |
---|---|
elsnc.1 | ⊢ A ∈ V |
Ref | Expression |
---|---|
elsnc | ⊢ (A ∈ {B} ↔ A = B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsnc.1 | . 2 ⊢ A ∈ V | |
2 | elsncg 3389 | . 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 1242 ∈ wcel 1390 Vcvv 2551 {csn 3367 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-sn 3373 |
This theorem is referenced by: sneqr 3522 onsucelsucexmid 4215 ordsoexmid 4240 opthprc 4334 dmsnm 4729 dmsnopg 4735 cnvcnvsn 4740 sniota 4837 fsn 5278 eusvobj2 5441 opelreal 6726 |
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