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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 3672 | . . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | 1 | biimpi 120 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
3 | 0ex 4145 | . 2 ⊢ ∅ ∈ V | |
4 | 2, 3 | eqeltrdi 2280 | 1 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ∅c0 3437 {csn 3607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-nul 4144 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-nul 3438 df-sn 3613 |
This theorem is referenced by: notnotsnex 4205 |
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