Theorem gruun 9943

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		uniiun 4793	. . 3 ⊢ ∪ {𝐴, 𝐵} = ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥
2		uniprg 4672	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
3	2	3adant1 1166	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
4	1, 3	syl5reqr 2876	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) = ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥)
5		simp1 1172	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 𝑈 ∈ Univ)
6		grupr 9934	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)
7		vex 3417	. . . . . . 7 ⊢ 𝑥 ∈ V
8	7	elpr 4420	. . . . . 6 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
9		eleq1a 2901	. . . . . . 7 ⊢ (𝐴 ∈ 𝑈 → (𝑥 = 𝐴 → 𝑥 ∈ 𝑈))
10		eleq1a 2901	. . . . . . 7 ⊢ (𝐵 ∈ 𝑈 → (𝑥 = 𝐵 → 𝑥 ∈ 𝑈))
11	9, 10	jaao 984	. . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑥 ∈ 𝑈))
12	8, 11	syl5bi 234	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝑥 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑈))
13	12	ralrimiv 3174	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
14	13	3adant1 1166	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
15		gruiun 9936	. . 3 ⊢ ((𝑈 ∈ Univ ∧ {𝐴, 𝐵} ∈ 𝑈 ∧ ∀𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
16	5, 6, 14, 15	syl3anc 1496	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
17	4, 16	eqeltrd 2906	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 386   ∨ wo 880   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∀wral 3117   ∪ cun 3796  {cpr 4399  ∪ cuni 4658  ∪ ciun 4740  Univcgru 9927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124  df-gru 9928

This theorem is referenced by:  gruxp  9944