Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pwidb | Structured version Visualization version GIF version |
Description: A class is an element of its powerclass if and only if it is a set. (Contributed by BJ, 31-Dec-2023.) |
Ref | Expression |
---|---|
pwidb | ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwidg 4554 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴) | |
2 | elex 3509 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐴 → 𝐴 ∈ V) | |
3 | 1, 2 | impbii 211 | 1 ⊢ (𝐴 ∈ V ↔ 𝐴 ∈ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2113 Vcvv 3491 𝒫 cpw 4532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-v 3493 df-in 3936 df-ss 3945 df-pw 4534 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |