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Mirrors > Home > MPE Home > Th. List > wunpw | Structured version Visualization version GIF version |
Description: A weak universe is closed under powerset. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wununi.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wununi.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wunpw | ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 4540 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | eleq1d 2897 | . 2 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ 𝑈 ↔ 𝒫 𝐴 ∈ 𝑈)) |
3 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
4 | iswun 10115 | . . . . 5 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
5 | 4 | ibi 268 | . . . 4 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))) |
6 | 5 | simp3d 1136 | . . 3 ⊢ (𝑈 ∈ WUni → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)) |
7 | simp2 1129 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → 𝒫 𝑥 ∈ 𝑈) | |
8 | 7 | ralimi 3160 | . . 3 ⊢ (∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑥 ∈ 𝑈 𝒫 𝑥 ∈ 𝑈) |
9 | 3, 6, 8 | 3syl 18 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 𝒫 𝑥 ∈ 𝑈) |
10 | wununi.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
11 | 2, 9, 10 | rspcdva 3624 | 1 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3016 ∀wral 3138 ∅c0 4290 𝒫 cpw 4537 {cpr 4561 ∪ cuni 4832 Tr wtr 5164 WUnicwun 10111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-in 3942 df-ss 3951 df-pw 4539 df-uni 4833 df-tr 5165 df-wun 10113 |
This theorem is referenced by: wunss 10123 wunr1om 10130 wunxp 10135 wunpm 10136 intwun 10146 r1wunlim 10148 wuncval2 10158 wuncn 10581 wunfunc 17159 wunnat 17216 catcoppccl 17358 catcfuccl 17359 catcxpccl 17447 ex-sategoelel 32566 |
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