Theorem grupr 10553

Description: A Grothendieck universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
grupr	⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)

Proof of Theorem grupr

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

Step	Hyp	Ref	Expression
1		elgrug 10548	. . . . . . 7 ⊢ (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))))
2	1	ibi 266	. . . . . 6 ⊢ (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈)))
3	2	simprd 496	. . . . 5 ⊢ (𝑈 ∈ Univ → ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))
4		preq2 4670	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
5	4	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝑥, 𝐵} ∈ 𝑈))
6	5	rspccv 3558	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 → (𝐵 ∈ 𝑈 → {𝑥, 𝐵} ∈ 𝑈))
7	6	3ad2ant2 1133	. . . . . . 7 ⊢ ((𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈) → (𝐵 ∈ 𝑈 → {𝑥, 𝐵} ∈ 𝑈))
8	7	com12 32	. . . . . 6 ⊢ (𝐵 ∈ 𝑈 → ((𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈) → {𝑥, 𝐵} ∈ 𝑈))
9	8	ralimdv 3109	. . . . 5 ⊢ (𝐵 ∈ 𝑈 → (∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈) → ∀𝑥 ∈ 𝑈 {𝑥, 𝐵} ∈ 𝑈))
10	3, 9	syl5com 31	. . . 4 ⊢ (𝑈 ∈ Univ → (𝐵 ∈ 𝑈 → ∀𝑥 ∈ 𝑈 {𝑥, 𝐵} ∈ 𝑈))
11		preq1 4669	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
12	11	eleq1d 2823	. . . . 5 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈))
13	12	rspccv 3558	. . . 4 ⊢ (∀𝑥 ∈ 𝑈 {𝑥, 𝐵} ∈ 𝑈 → (𝐴 ∈ 𝑈 → {𝐴, 𝐵} ∈ 𝑈))
14	10, 13	syl6 35	. . 3 ⊢ (𝑈 ∈ Univ → (𝐵 ∈ 𝑈 → (𝐴 ∈ 𝑈 → {𝐴, 𝐵} ∈ 𝑈)))
15	14	com23 86	. 2 ⊢ (𝑈 ∈ Univ → (𝐴 ∈ 𝑈 → (𝐵 ∈ 𝑈 → {𝐴, 𝐵} ∈ 𝑈)))
16	15	3imp 1110	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  𝒫 cpw 4533  {cpr 4563  ∪ cuni 4839  Tr wtr 5191  ran crn 5590  (class class class)co 7275   ↑_m cmap 8615  Univcgru 10546

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-tr 5192  df-iota 6391  df-fv 6441  df-ov 7278  df-gru 10547

This theorem is referenced by:  grusn  10560  gruop  10561  gruun  10562  gruwun  10569  intgru  10570