Theorem grupw 10718

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 10715	. . . . 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 267	. . . 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 495	. . 3 ⊢ (𝑈 ∈ Univ → ∀𝑦 ∈ 𝑈 (𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈))
4		simp1 1137	. . . 4 ⊢ ((𝒫 𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 {𝑦, 𝑥} ∈ 𝑈 ∧ ∀𝑥 ∈ (𝑈 ↑m 𝑦)∪ ran 𝑥 ∈ 𝑈) → 𝒫 𝑦 ∈ 𝑈)
5	4	ralimi 3074	. . 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 4555	. . . . 5 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
7	6	eleq1d 2821	. . . 4 ⊢ (𝑦 = 𝐴 → (𝒫 𝑦 ∈ 𝑈 ↔ 𝒫 𝐴 ∈ 𝑈))
8	7	rspccv 3561	. . 3 ⊢ (∀𝑦 ∈ 𝑈 𝒫 𝑦 ∈ 𝑈 → (𝐴 ∈ 𝑈 → 𝒫 𝐴 ∈ 𝑈))
9	3, 5, 8	3syl 18	. 2 ⊢ (𝑈 ∈ Univ → (𝐴 ∈ 𝑈 → 𝒫 𝐴 ∈ 𝑈))
10	9	imp 406	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  𝒫 cpw 4541  {cpr 4569  ∪ cuni 4850  Tr wtr 5192  ran crn 5632  (class class class)co 7367   ↑_m cmap 8773  Univcgru 10713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-tr 5193  df-iota 6454  df-fv 6506  df-ov 7370  df-gru 10714

This theorem is referenced by:  gruss  10719  grurn  10724  gruxp  10730  grumap  10731  gruwun  10736  intgru  10737  gruina  10741  grur1a  10742  grur1cld  44659  grumnudlem  44712