Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > pwexb | GIF version |
Description: The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
pwexb | ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexb 4445 | . 2 ⊢ (𝒫 𝐴 ∈ V ↔ ∪ 𝒫 𝐴 ∈ V) | |
2 | unipw 4189 | . . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
3 | 2 | eleq1i 2230 | . 2 ⊢ (∪ 𝒫 𝐴 ∈ V ↔ 𝐴 ∈ V) |
4 | 1, 3 | bitr2i 184 | 1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 2135 Vcvv 2721 𝒫 cpw 3553 ∪ cuni 3783 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-uni 3784 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |