Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sselpwd | Structured version Visualization version GIF version |
Description: Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.) |
Ref | Expression |
---|---|
sselpwd.1 | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
sselpwd.2 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
sselpwd | ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sselpwd.2 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sselpwd.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | 2, 1 | ssexd 4733 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
4 | elpwg 4116 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
6 | 1, 5 | mpbird 246 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-pw 4110 |
This theorem is referenced by: fin1a2lem7 9111 ustssel 21819 ldsysgenld 29550 ldgenpisyslem1 29553 rfovcnvf1od 37318 fsovrfovd 37323 fsovfd 37326 fsovcnvlem 37327 ntrclsrcomplex 37353 clsk3nimkb 37358 clsk1indlem3 37361 clsk1indlem4 37362 clsk1indlem1 37363 ntrclsiso 37385 ntrclskb 37387 ntrclsk3 37388 ntrclsk13 37389 ntrneircomplex 37392 ntrneik3 37414 ntrneix3 37415 ntrneik13 37416 ntrneix13 37417 clsneircomplex 37421 clsneiel1 37426 neicvgrcomplex 37431 neicvgel1 37437 ovolsplit 38881 |
Copyright terms: Public domain | W3C validator |