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Theorem wunpw 9730
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 4298 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
21eleq1d 2834 . 2 (𝑥 = 𝐴 → (𝒫 𝑥𝑈 ↔ 𝒫 𝐴𝑈))
3 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
4 iswun 9727 . . . . 5 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
54ibi 256 . . . 4 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
65simp3d 1137 . . 3 (𝑈 ∈ WUni → ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
7 simp2 1130 . . . 4 (( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → 𝒫 𝑥𝑈)
87ralimi 3100 . . 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 3464 1 (𝜑 → 𝒫 𝐴𝑈)
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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