![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > vpwex | GIF version |
Description: Power set axiom: the powerclass of a set is a set. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Revised to prove pwexg 4112 from vpwex 4111. (Revised by BJ, 10-Aug-2022.) |
Ref | Expression |
---|---|
vpwex | ⊢ 𝒫 𝑥 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pw 3517 | . 2 ⊢ 𝒫 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑥} | |
2 | axpow2 4108 | . . . . 5 ⊢ ∃𝑧∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝑧) | |
3 | 2 | bm1.3ii 4057 | . . . 4 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥) |
4 | abeq2 2249 | . . . . 5 ⊢ (𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥)) | |
5 | 4 | exbii 1585 | . . . 4 ⊢ (∃𝑧 𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥} ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥)) |
6 | 3, 5 | mpbir 145 | . . 3 ⊢ ∃𝑧 𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥} |
7 | 6 | issetri 2698 | . 2 ⊢ {𝑦 ∣ 𝑦 ⊆ 𝑥} ∈ V |
8 | 1, 7 | eqeltri 2213 | 1 ⊢ 𝒫 𝑥 ∈ V |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1330 = wceq 1332 ∃wex 1469 ∈ wcel 1481 {cab 2126 Vcvv 2689 ⊆ wss 3076 𝒫 cpw 3515 |
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-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-v 2691 df-in 3082 df-ss 3089 df-pw 3517 |
This theorem is referenced by: pwexg 4112 pwnex 4378 istopon 12219 dmtopon 12229 tgdom 12280 |
Copyright terms: Public domain | W3C validator |