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Mirrors > Home > ILE Home > Th. List > snelpw | GIF version |
Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.) |
Ref | Expression |
---|---|
snelpw.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
snelpw | ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snelpw.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | snss 3644 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵) |
3 | 1 | snex 4104 | . . 3 ⊢ {𝐴} ∈ V |
4 | 3 | elpw 3511 | . 2 ⊢ ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵) |
5 | 2, 4 | bitr4i 186 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1480 Vcvv 2681 ⊆ wss 3066 𝒫 cpw 3505 {csn 3522 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 |
This theorem is referenced by: (None) |
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