Theorem grupw 10893

Description: A Grothendieck universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
grupw	⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ∈ 𝑈)

Proof of Theorem grupw

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

Step	Hyp	Ref	Expression
1		elgrug 10890	. . . . 5 ⊢ (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈))))
2	1	ibi 270	. . . 4 ⊢ (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈)))
3	2	simprd 501	. . 3 ⊢ (𝑈 ∈ Univ → ∀𝑦 ∈ 𝑈 (𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈))
4		simp1 1154	. . . 4 ⊢ ((𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈) → 𝒫 𝑦 ∈ 𝑈)
5	4	ralimi 3103	. . 3 ⊢ (∀𝑦 ∈ 𝑈 (𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈) → ∀𝑦 ∈ 𝑈 𝒫 𝑦 ∈ 𝑈)
6		pweq 4574	. . . . 5 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
7	6	eleq1d 2849	. . . 4 ⊢ (𝑦 = 𝐴 → (𝒫 𝑦 ∈ 𝑈 ↔ 𝒫 𝐴 ∈ 𝑈))
8	7	rspccv 3577	. . 3 ⊢ (∀𝑦 ∈ 𝑈 𝒫 𝑦 ∈ 𝑈 → (𝐴 ∈ 𝑈 → 𝒫 𝐴 ∈ 𝑈))
9	3, 5, 8	3syl 19	. 2 ⊢ (𝑈 ∈ Univ → (𝐴 ∈ 𝑈 → 𝒫 𝐴 ∈ 𝑈))
10	9	imp 412	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∀wral 3080  𝒫 cpw 4560  {cpr 4590  ∪ cuni 4870  Tr wtr 5215  ran crn 5656  (class class class)co 7430   ↑_m cmap 8861  Univcgru 10888

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-tr 5216  df-iota 6501  df-fv 6555  df-ov 7433  df-gru 10889

This theorem is used by:  gruss  10894  grurn  10899  gruxp  10905  grumap  10906  gruwun  10911  intgru  10912  gruina  10916  grur1a  10917  grur1cld  45373  grumnudlem  45412