Theorem wunpw 10660

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 4577	. . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2	1	eleq1d 2813	. 2 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ 𝑈 ↔ 𝒫 𝐴 ∈ 𝑈))
3		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
4		iswun 10657	. . . . 5 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))
5	4	ibi 267	. . . 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 3066	. . 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 3589	1 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∅c0 4296  𝒫 cpw 4563  {cpr 4591  ∪ cuni 4871  Tr wtr 5214  WUnicwun 10653

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-v 3449  df-ss 3931  df-pw 4565  df-uni 4872  df-tr 5215  df-wun 10655

This theorem is referenced by:  wunss  10665  wunr1om  10672  wunxp  10677  wunpm  10678  intwun  10688  r1wunlim  10690  wuncval2  10700  wuncn  11123  wunfunc  17863  wunnat  17921  catcoppccl  18079  catcfuccl  18080  catcxpccl  18168  ex-sategoelel  35408