Theorem wunpw 10701

Description: A weak universe is closed under powerset. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wununi.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wununi.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Assertion

Ref	Expression
wunpw	⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈)

Proof of Theorem wunpw

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4616	. . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2	1	eleq1d 2818	. 2 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ 𝑈 ↔ 𝒫 𝐴 ∈ 𝑈))
3		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
4		iswun 10698	. . . . 5 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))
5	4	ibi 266	. . . 4 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))
6	5	simp3d 1144	. . 3 ⊢ (𝑈 ∈ WUni → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))
7		simp2 1137	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → 𝒫 𝑥 ∈ 𝑈)
8	7	ralimi 3083	. . 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 3613	1 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∅c0 4322  𝒫 cpw 4602  {cpr 4630  ∪ cuni 4908  Tr wtr 5265  WUnicwun 10694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3476  df-in 3955  df-ss 3965  df-pw 4604  df-uni 4909  df-tr 5266  df-wun 10696

This theorem is referenced by:  wunss  10706  wunr1om  10713  wunxp  10718  wunpm  10719  intwun  10729  r1wunlim  10731  wuncval2  10741  wuncn  11164  wunfunc  17848  wunfuncOLD  17849  wunnat  17906  wunnatOLD  17907  catcoppccl  18066  catcoppcclOLD  18067  catcfuccl  18068  catcfucclOLD  18069  catcxpccl  18158  catcxpcclOLD  18159  ex-sategoelel  34407