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Theorem wunpw 10651
Description: A weak universe is closed under powerset. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1 (𝜑𝑈 ∈ WUni)
wununi.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wunpw (𝜑 → 𝒫 𝐴𝑈)

Proof of Theorem wunpw
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pweq 4578 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
21eleq1d 2819 . 2 (𝑥 = 𝐴 → (𝒫 𝑥𝑈 ↔ 𝒫 𝐴𝑈))
3 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
4 iswun 10648 . . . . 5 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
54ibi 267 . . . 4 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
65simp3d 1145 . . 3 (𝑈 ∈ WUni → ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
7 simp2 1138 . . . 4 (( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → 𝒫 𝑥𝑈)
87ralimi 3083 . . 3 (∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑥𝑈 𝒫 𝑥𝑈)
93, 6, 83syl 18 . 2 (𝜑 → ∀𝑥𝑈 𝒫 𝑥𝑈)
10 wununi.2 . 2 (𝜑𝐴𝑈)
112, 9, 10rspcdva 3584 1 (𝜑 → 𝒫 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wral 3061  c0 4286  𝒫 cpw 4564  {cpr 4592   cuni 4869  Tr wtr 5226  WUnicwun 10644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-v 3449  df-in 3921  df-ss 3931  df-pw 4566  df-uni 4870  df-tr 5227  df-wun 10646
This theorem is referenced by:  wunss  10656  wunr1om  10663  wunxp  10668  wunpm  10669  intwun  10679  r1wunlim  10681  wuncval2  10691  wuncn  11114  wunfunc  17793  wunfuncOLD  17794  wunnat  17851  wunnatOLD  17852  catcoppccl  18011  catcoppcclOLD  18012  catcfuccl  18013  catcfucclOLD  18014  catcxpccl  18103  catcxpcclOLD  18104  ex-sategoelel  34079
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