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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 4558 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | eleq1d 2900 | . 2 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ 𝑈 ↔ 𝒫 𝐴 ∈ 𝑈)) |
3 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
4 | iswun 10129 | . . . . 5 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
5 | 4 | ibi 269 | . . . 4 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))) |
6 | 5 | simp3d 1140 | . . 3 ⊢ (𝑈 ∈ WUni → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)) |
7 | simp2 1133 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → 𝒫 𝑥 ∈ 𝑈) | |
8 | 7 | ralimi 3163 | . . 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 3628 | 1 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∀wral 3141 ∅c0 4294 𝒫 cpw 4542 {cpr 4572 ∪ cuni 4841 Tr wtr 5175 WUnicwun 10125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-v 3499 df-in 3946 df-ss 3955 df-pw 4544 df-uni 4842 df-tr 5176 df-wun 10127 |
This theorem is referenced by: wunss 10137 wunr1om 10144 wunxp 10149 wunpm 10150 intwun 10160 r1wunlim 10162 wuncval2 10172 wuncn 10595 wunfunc 17172 wunnat 17229 catcoppccl 17371 catcfuccl 17372 catcxpccl 17460 ex-sategoelel 32672 |
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