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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 3672 | . 2 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
2 | elex 2700 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | snexg 4116 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
4 | elpwg 3523 | . . 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 1481 Vcvv 2689 ⊆ wss 3076 𝒫 cpw 3515 {csn 3532 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 |
This theorem is referenced by: unipw 4147 infpwfidom 7071 txdis 12485 txdis1cn 12486 |
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