Theorem grupr 9657

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 9652	. . . . . . 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 256	. . . . . 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 478	. . . . 5 ⊢ (𝑈 ∈ Univ → ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑈))
4		preq2 4301	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
5	4	eleq1d 2715	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝑥, 𝐵} ∈ 𝑈))
6	5	rspccv 3337	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 → (𝐵 ∈ 𝑈 → {𝑥, 𝐵} ∈ 𝑈))
7	6	3ad2ant2 1103	. . . . . . 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 2992	. . . . 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 4300	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
12	11	eleq1d 2715	. . . . 5 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈))
13	12	rspccv 3337	. . . 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 1275	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  𝒫 cpw 4191  {cpr 4212  ∪ cuni 4468  Tr wtr 4785  ran crn 5144  (class class class)co 6690   ↑_𝑚 cmap 7899  Univcgru 9650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-tr 4786  df-iota 5889  df-fv 5934  df-ov 6693  df-gru 9651

This theorem is referenced by:  grusn  9664  gruop  9665  gruun  9666  gruwun  9673  intgru  9674