Theorem gruun 10875

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 4947	. . . 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 5081	. . 3 ⊢ ∪ {𝐴, 𝐵} = ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥
4	2, 3	eqtr3di 2795	. 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 10866	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)
7		vex 3492	. . . . . . 7 ⊢ 𝑥 ∈ V
8	7	elpr 4672	. . . . . 6 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
9		eleq1a 2839	. . . . . . 7 ⊢ (𝐴 ∈ 𝑈 → (𝑥 = 𝐴 → 𝑥 ∈ 𝑈))
10		eleq1a 2839	. . . . . . 7 ⊢ (𝐵 ∈ 𝑈 → (𝑥 = 𝐵 → 𝑥 ∈ 𝑈))
11	9, 10	jaao 955	. . . . . 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 3151	. . . 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 10868	. . 3 ⊢ ((𝑈 ∈ Univ ∧ {𝐴, 𝐵} ∈ 𝑈 ∧ ∀𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
16	5, 6, 14, 15	syl3anc 1371	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
17	4, 16	eqeltrd 2844	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∪ cun 3974  {cpr 4650  ∪ cuni 4931  ∪ ciun 5015  Univcgru 10859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-gru 10860

This theorem is referenced by:  gruxp  10876  grusucd  44199  grumnudlem  44254