![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > elpwb | GIF version |
Description: Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.) |
Ref | Expression |
---|---|
elpwb | ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2763 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ∈ V) | |
2 | elpwg 3598 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
3 | 1, 2 | biadan2 456 | 1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 𝒫 cpw 3590 |
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-ext 2171 |
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 |
This theorem is referenced by: elpwpw 3988 elpwpwel 4493 |
Copyright terms: Public domain | W3C validator |