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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 4180 from vpwex 4179. (Revised by BJ, 10-Aug-2022.) |
Ref | Expression |
---|---|
vpwex | ⊢ 𝒫 𝑥 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pw 3577 | . 2 ⊢ 𝒫 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑥} | |
2 | axpow2 4176 | . . . . 5 ⊢ ∃𝑧∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝑧) | |
3 | 2 | bm1.3ii 4124 | . . . 4 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥) |
4 | abeq2 2286 | . . . . 5 ⊢ (𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥)) | |
5 | 4 | exbii 1605 | . . . 4 ⊢ (∃𝑧 𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥} ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥)) |
6 | 3, 5 | mpbir 146 | . . 3 ⊢ ∃𝑧 𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑥} |
7 | 6 | issetri 2746 | . 2 ⊢ {𝑦 ∣ 𝑦 ⊆ 𝑥} ∈ V |
8 | 1, 7 | eqeltri 2250 | 1 ⊢ 𝒫 𝑥 ∈ V |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∀wal 1351 = wceq 1353 ∃wex 1492 ∈ wcel 2148 {cab 2163 Vcvv 2737 ⊆ wss 3129 𝒫 cpw 3575 |
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-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-v 2739 df-in 3135 df-ss 3142 df-pw 3577 |
This theorem is referenced by: pwexg 4180 pwnex 4449 istopon 13483 dmtopon 13493 tgdom 13542 |
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