Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > elpwi2 | GIF version |
Description: Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.) |
Ref | Expression |
---|---|
elpwi2.1 | ⊢ 𝐵 ∈ 𝑉 |
elpwi2.2 | ⊢ 𝐴 ⊆ 𝐵 |
Ref | Expression |
---|---|
elpwi2 | ⊢ 𝐴 ∈ 𝒫 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi2.2 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | elpwi2.1 | . . . 4 ⊢ 𝐵 ∈ 𝑉 | |
3 | 2 | elexi 2742 | . . 3 ⊢ 𝐵 ∈ V |
4 | 3 | elpw2 4143 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) |
5 | 1, 4 | mpbir 145 | 1 ⊢ 𝐴 ∈ 𝒫 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 ⊆ wss 3121 𝒫 cpw 3566 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-sep 4107 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-pw 3568 |
This theorem is referenced by: canth 5807 |
Copyright terms: Public domain | W3C validator |