![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > vpwex | Structured version Visualization version 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 5384 from vpwex 5383. (Revised by BJ, 10-Aug-2022.) |
Ref | Expression |
---|---|
vpwex | ⊢ 𝒫 𝑥 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pw 4607 | . 2 ⊢ 𝒫 𝑥 = {𝑤 ∣ 𝑤 ⊆ 𝑥} | |
2 | axpow2 5373 | . . . . 5 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦) | |
3 | 2 | sepexi 5307 | . . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥) |
4 | sseq1 4021 | . . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑤 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑥)) | |
5 | 4 | eqabbw 2813 | . . . . 5 ⊢ (𝑦 = {𝑤 ∣ 𝑤 ⊆ 𝑥} ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥)) |
6 | 5 | exbii 1845 | . . . 4 ⊢ (∃𝑦 𝑦 = {𝑤 ∣ 𝑤 ⊆ 𝑥} ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥)) |
7 | 3, 6 | mpbir 231 | . . 3 ⊢ ∃𝑦 𝑦 = {𝑤 ∣ 𝑤 ⊆ 𝑥} |
8 | 7 | issetri 3497 | . 2 ⊢ {𝑤 ∣ 𝑤 ⊆ 𝑥} ∈ V |
9 | 1, 8 | eqeltri 2835 | 1 ⊢ 𝒫 𝑥 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∀wal 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 Vcvv 3478 ⊆ wss 3963 𝒫 cpw 4605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-pow 5371 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-pw 4607 |
This theorem is referenced by: pwexg 5384 pwnex 7778 inf3lem7 9672 dfac8 10174 dfac13 10181 ackbij1lem8 10264 dominf 10483 numthcor 10532 dominfac 10611 intwun 10773 wunex2 10776 eltsk2g 10789 inttsk 10812 tskcard 10819 intgru 10852 gruina 10856 axgroth6 10866 ismre 17635 fnmre 17636 mreacs 17703 isacs5lem 18603 pmtrfval 19483 istopon 22934 dmtopon 22945 tgdom 23001 isfbas 23853 bj-snglex 36956 exrecfnpw 37364 pwinfi 43554 ntrrn 44112 ntrf 44113 dssmapntrcls 44118 vsetrec 48934 pgindnf 48947 |
Copyright terms: Public domain | W3C validator |