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Mirrors > Home > ILE Home > Th. List > snexgOLD | 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. This is a special case of snexg 3927 and new proofs should use snexg 3927 instead. (Contributed by Jim Kingdon, 26-Jan-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of snexg 3927 and then remove it. |
Ref | Expression |
---|---|
snexgOLD | ⊢ (A ∈ V → {A} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 3924 | . 2 ⊢ (A ∈ V → 𝒫 A ∈ V) | |
2 | snsspw 3526 | . . 3 ⊢ {A} ⊆ 𝒫 A | |
3 | ssexg 3887 | . . 3 ⊢ (({A} ⊆ 𝒫 A ∧ 𝒫 A ∈ V) → {A} ∈ V) | |
4 | 2, 3 | mpan 400 | . 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 1390 Vcvv 2551 ⊆ wss 2911 𝒫 cpw 3351 {csn 3367 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 |
This theorem is referenced by: snelpwi 3939 snelpw 3940 rext 3942 sspwb 3943 intid 3951 euabex 3952 mss 3953 exss 3954 opexgOLD 3956 opi1 3960 opm 3962 opeqsn 3980 opeqpr 3981 uniop 3983 snnex 4147 op1stb 4175 op1stbg 4176 sucexb 4189 dtruex 4237 relop 4429 elxp4 4751 elxp5 4752 funopg 4877 1stvalg 5711 2ndvalg 5712 fo1st 5726 fo2nd 5727 |
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