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Mirrors > Home > ILE Home > Th. List > snprc | GIF version |
Description: The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
snprc | ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | velsn 3415 | . . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
2 | 1 | exbii 1536 | . . 3 ⊢ (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴) |
3 | 2 | notbii 626 | . 2 ⊢ (¬ ∃𝑥 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 = 𝐴) |
4 | eq0 3266 | . . 3 ⊢ ({𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝐴}) | |
5 | alnex 1428 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴}) | |
6 | 4, 5 | bitri 182 | . 2 ⊢ ({𝐴} = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴}) |
7 | isset 2605 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
8 | 7 | notbii 626 | . 2 ⊢ (¬ 𝐴 ∈ V ↔ ¬ ∃𝑥 𝑥 = 𝐴) |
9 | 3, 6, 8 | 3bitr4ri 211 | 1 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 103 ∀wal 1282 = wceq 1284 ∃wex 1421 ∈ wcel 1433 Vcvv 2601 ∅c0 3251 {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-in1 576 ax-in2 577 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-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 df-nul 3252 df-sn 3404 |
This theorem is referenced by: prprc1 3500 prprc 3502 snexprc 3958 sucprc 4167 snnen2oprc 6346 unsnfidcex 6385 |
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