![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3751 | . 2 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
2 | elex 2763 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | snexg 4202 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
4 | elpwg 3598 | . . 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 167 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 𝒫 cpw 3590 {csn 3607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 |
This theorem is referenced by: unipw 4235 infpwfidom 7227 txdis 14237 txdis1cn 14238 |
Copyright terms: Public domain | W3C validator |