Theorem grupr 10694

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 10689	. . . . . . 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 267	. . . . . 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 495	. . . . 5 ⊢ (𝑈 ∈ Univ → ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))
4		preq2 4686	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
5	4	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝑥, 𝐵} ∈ 𝑈))
6	5	rspccv 3569	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 → (𝐵 ∈ 𝑈 → {𝑥, 𝐵} ∈ 𝑈))
7	6	3ad2ant2 1134	. . . . . . 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 3146	. . . . 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 4685	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
12	11	eleq1d 2816	. . . . 5 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈))
13	12	rspccv 3569	. . . 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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  𝒫 cpw 4549  {cpr 4577  ∪ cuni 4858  Tr wtr 5200  ran crn 5620  (class class class)co 7352   ↑_m cmap 8756  Univcgru 10687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-tr 5201  df-iota 6443  df-fv 6495  df-ov 7355  df-gru 10688

This theorem is referenced by:  grusn  10701  gruop  10702  gruun  10703  gruwun  10710  intgru  10711