Theorem gruun 10825

Description: A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
gruun	⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) ∈ 𝑈)

Proof of Theorem gruun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniprg 4904	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
2	1	3adant1 1130	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
3		uniiun 5039	. . 3 ⊢ ∪ {𝐴, 𝐵} = ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥
4	2, 3	eqtr3di 2786	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) = ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥)
5		simp1 1136	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 𝑈 ∈ Univ)
6		grupr 10816	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)
7		vex 3468	. . . . . . 7 ⊢ 𝑥 ∈ V
8	7	elpr 4631	. . . . . 6 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
9		eleq1a 2830	. . . . . . 7 ⊢ (𝐴 ∈ 𝑈 → (𝑥 = 𝐴 → 𝑥 ∈ 𝑈))
10		eleq1a 2830	. . . . . . 7 ⊢ (𝐵 ∈ 𝑈 → (𝑥 = 𝐵 → 𝑥 ∈ 𝑈))
11	9, 10	jaao 956	. . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑥 ∈ 𝑈))
12	8, 11	biimtrid 242	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝑥 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑈))
13	12	ralrimiv 3132	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
14	13	3adant1 1130	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
15		gruiun 10818	. . 3 ⊢ ((𝑈 ∈ Univ ∧ {𝐴, 𝐵} ∈ 𝑈 ∧ ∀𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
16	5, 6, 14, 15	syl3anc 1373	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
17	4, 16	eqeltrd 2835	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052   ∪ cun 3929  {cpr 4608  ∪ cuni 4888  ∪ ciun 4972  Univcgru 10809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8847  df-gru 10810

This theorem is referenced by:  gruxp  10826  grusucd  44221  grumnudlem  44276