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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 3588 | . . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | 1 | biimpi 119 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
3 | 0ex 4055 | . 2 ⊢ ∅ ∈ V | |
4 | 2, 3 | eqeltrdi 2230 | 1 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∅c0 3363 {csn 3527 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-nul 4054 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 df-nul 3364 df-sn 3533 |
This theorem is referenced by: notnotsnex 4111 |
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