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Mirrors > Home > ILE Home > Th. List > snelpwi | GIF version |
Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.) |
Ref | Expression |
---|---|
snelpwi | ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 3722 | . 2 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
2 | elex 2741 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | snexg 4168 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
4 | elpwg 3572 | . . 3 ⊢ ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)) | |
5 | 2, 3, 4 | 3syl 17 | . 2 ⊢ (𝐴 ∈ 𝐵 → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)) |
6 | 1, 5 | mpbird 166 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 2141 Vcvv 2730 ⊆ wss 3121 𝒫 cpw 3564 {csn 3581 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 |
This theorem is referenced by: unipw 4200 infpwfidom 7162 txdis 12992 txdis1cn 12993 |
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