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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 3702 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵) |
3 | 1 | snex 4164 | . . 3 ⊢ {𝐴} ∈ V |
4 | 3 | elpw 3565 | . 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 2136 Vcvv 2726 ⊆ wss 3116 𝒫 cpw 3559 {csn 3576 |
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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 |
This theorem depends on definitions: df-bi 116 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-v 2728 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 |
This theorem is referenced by: (None) |
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