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Theorem wunpw 9408
 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 wununi.2 . 2 (𝜑𝐴𝑈)
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 iswun 9405 . . . . 5 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
43ibi 255 . . . 4 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
54simp3d 1068 . . 3 (𝑈 ∈ WUni → ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
6 simp2 1055 . . . 4 (( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → 𝒫 𝑥𝑈)
76ralimi 2936 . . 3 (∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑥𝑈 𝒫 𝑥𝑈)
82, 5, 73syl 18 . 2 (𝜑 → ∀𝑥𝑈 𝒫 𝑥𝑈)
9 pweq 4111 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
109eleq1d 2672 . . 3 (𝑥 = 𝐴 → (𝒫 𝑥𝑈 ↔ 𝒫 𝐴𝑈))
1110rspcv 3278 . 2 (𝐴𝑈 → (∀𝑥𝑈 𝒫 𝑥𝑈 → 𝒫 𝐴𝑈))
121, 8, 11sylc 63 1 (𝜑 → 𝒫 𝐴𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∅c0 3874  𝒫 cpw 4108  {cpr 4127  ∪ cuni 4372  Tr wtr 4680  WUnicwun 9401 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373  df-tr 4681  df-wun 9403 This theorem is referenced by:  wunss  9413  wunr1om  9420  wunxp  9425  wunpm  9426  intwun  9436  r1wunlim  9438  wuncval2  9448  wuncn  9870  wunfunc  16382  wunnat  16439  catcoppccl  16581  catcfuccl  16582  catcxpccl  16670
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