Theorem grupr 9876

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 9871	. . . . . . 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 258	. . . . . 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 489	. . . . 5 ⊢ (𝑈 ∈ Univ → ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑈))
4		preq2 4426	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵})
5	4	eleq1d 2829	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝑥, 𝐵} ∈ 𝑈))
6	5	rspccv 3459	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 → (𝐵 ∈ 𝑈 → {𝑥, 𝐵} ∈ 𝑈))
7	6	3ad2ant2 1164	. . . . . . 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 3110	. . . . 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 4425	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
12	11	eleq1d 2829	. . . . 5 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈))
13	12	rspccv 3459	. . . 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 1137	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155  ∀wral 3055  𝒫 cpw 4317  {cpr 4338  ∪ cuni 4596  Tr wtr 4913  ran crn 5280  (class class class)co 6846   ↑_𝑚 cmap 8064  Univcgru 9869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-tr 4914  df-iota 6033  df-fv 6078  df-ov 6849  df-gru 9870

This theorem is referenced by:  grusn  9883  gruop  9884  gruun  9885  gruwun  9892  intgru  9893