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Mirrors > Home > ILE Home > Th. List > snexg | GIF version |
Description: A singleton whose element exists 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 |
---|---|
snexg | ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 4021 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
2 | snsspw 3614 | . . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴 | |
3 | ssexg 3984 | . . 3 ⊢ (({𝐴} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝐴} ∈ V) | |
4 | 2, 3 | mpan 416 | . 2 ⊢ (𝒫 𝐴 ∈ V → {𝐴} ∈ V) |
5 | 1, 4 | syl 14 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1439 Vcvv 2620 ⊆ wss 3000 𝒫 cpw 3433 {csn 3450 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 |
This theorem is referenced by: snex 4026 notnotsnex 4028 snelpwi 4048 opexg 4064 opm 4070 tpexg 4279 op1stbg 4314 sucexb 4327 elxp4 4931 elxp5 4932 opabex3d 5906 opabex3 5907 1stvalg 5927 2ndvalg 5928 mpt2exxg 5991 cnvf1o 6004 brtpos2 6030 tfr0dm 6101 tfrlemisucaccv 6104 tfrlemibxssdm 6106 tfrlemibfn 6107 tfr1onlemsucaccv 6120 tfr1onlembxssdm 6122 tfr1onlembfn 6123 tfrcllemsucaccv 6133 tfrcllembxssdm 6135 tfrcllembfn 6136 fvdiagfn 6464 ixpsnf1o 6507 mapsnf1o 6508 xpsnen2g 6599 zfz1isolem1 10306 climconst2 10740 setsvalg 11585 setsex 11587 setsslid 11605 strle1g 11645 1strbas 11654 |
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