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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 4298 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | eleq1d 2834 | . 2 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ 𝑈 ↔ 𝒫 𝐴 ∈ 𝑈)) |
3 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
4 | iswun 9727 | . . . . 5 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
5 | 4 | ibi 256 | . . . 4 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))) |
6 | 5 | simp3d 1137 | . . 3 ⊢ (𝑈 ∈ WUni → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)) |
7 | simp2 1130 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → 𝒫 𝑥 ∈ 𝑈) | |
8 | 7 | ralimi 3100 | . . 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 3464 | 1 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1070 = wceq 1630 ∈ wcel 2144 ≠ wne 2942 ∀wral 3060 ∅c0 4061 𝒫 cpw 4295 {cpr 4316 ∪ cuni 4572 Tr wtr 4884 WUnicwun 9723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-v 3351 df-in 3728 df-ss 3735 df-pw 4297 df-uni 4573 df-tr 4885 df-wun 9725 |
This theorem is referenced by: wunss 9735 wunr1om 9742 wunxp 9747 wunpm 9748 intwun 9758 r1wunlim 9760 wuncval2 9770 wuncn 10192 wunfunc 16765 wunnat 16822 catcoppccl 16964 catcfuccl 16965 catcxpccl 17054 |
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