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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 4141 from vpwex 4140. (Revised by BJ, 10-Aug-2022.) |
Ref | Expression |
---|---|
vpwex | ⊢ 𝒫 𝑥 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pw 3545 | . 2 ⊢ 𝒫 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑥} | |
2 | axpow2 4137 | . . . . 5 ⊢ ∃𝑧∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝑧) | |
3 | 2 | bm1.3ii 4085 | . . . 4 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ⊆ 𝑥) |
4 | abeq2 2266 | . . . . 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 2721 | . 2 ⊢ {𝑦 ∣ 𝑦 ⊆ 𝑥} ∈ V |
8 | 1, 7 | eqeltri 2230 | 1 ⊢ 𝒫 𝑥 ∈ V |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1333 = wceq 1335 ∃wex 1472 ∈ wcel 2128 {cab 2143 Vcvv 2712 ⊆ wss 3102 𝒫 cpw 3543 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-11 1486 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-v 2714 df-in 3108 df-ss 3115 df-pw 3545 |
This theorem is referenced by: pwexg 4141 pwnex 4409 istopon 12422 dmtopon 12432 tgdom 12483 |
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