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Mirrors > Home > ILE Home > Th. List > snexprc | GIF version |
Description: A singleton whose element is a proper class is a set. The ¬ 𝐴 ∈ V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.) |
Ref | Expression |
---|---|
snexprc | ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snprc 3505 | . . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | 1 | biimpi 118 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
3 | 0ex 3964 | . 2 ⊢ ∅ ∈ V | |
4 | 2, 3 | syl6eqel 2178 | 1 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1289 ∈ wcel 1438 Vcvv 2619 ∅c0 3286 {csn 3444 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-nul 3963 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-dif 3001 df-nul 3287 df-sn 3450 |
This theorem is referenced by: notnotsnex 4020 |
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