![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > elpwid | GIF version |
Description: An element of a power class is a subclass. Deduction form of elpwi 3583. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
elpwid.1 | ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) |
Ref | Expression |
---|---|
elpwid | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwid.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) | |
2 | elpwi 3583 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ⊆ wss 3129 𝒫 cpw 3574 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-ss 3142 df-pw 3576 |
This theorem is referenced by: fopwdom 6830 ssenen 6845 fival 6963 fiuni 6971 3nelsucpw1 7227 elnp1st2nd 7466 ixxssxr 9887 elfzoelz 10133 restid2 12645 epttop 13257 neiss2 13309 blssm 13588 blin2 13599 cncfrss 13729 cncfrss2 13730 pwle2 14404 |
Copyright terms: Public domain | W3C validator |