Theorem gruun 10884

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 4883	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
2	1	3adant1 1148	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
3		uniiun 5017	. . 3 ⊢ ∪ {𝐴, 𝐵} = ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥
4	2, 3	eqtr3di 2811	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) = ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥)
5		simp1 1154	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 𝑈 ∈ Univ)
6		grupr 10875	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)
7		vex 3455	. . . . . . 7 ⊢ 𝑥 ∈ V
8	7	elpr 4609	. . . . . 6 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
9		eleq1a 2856	. . . . . . 7 ⊢ (𝐴 ∈ 𝑈 → (𝑥 = 𝐴 → 𝑥 ∈ 𝑈))
10		eleq1a 2856	. . . . . . 7 ⊢ (𝐵 ∈ 𝑈 → (𝑥 = 𝐵 → 𝑥 ∈ 𝑈))
11	9, 10	jaao 969	. . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑥 ∈ 𝑈))
12	8, 11	biimtrid 245	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝑥 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑈))
13	12	ralrimiv 3154	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
14	13	3adant1 1148	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
15		gruiun 10877	. . 3 ⊢ ((𝑈 ∈ Univ ∧ {𝐴, 𝐵} ∈ 𝑈 ∧ ∀𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
16	5, 6, 14, 15	syl3anc 1398	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝑥 ∈ 𝑈)
17	4, 16	eqeltrd 2861	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ∪ cun 3897  {cpr 4586  ∪ cuni 4867  ∪ ciun 4951  Univcgru 10868

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-map 8842  df-gru 10869

This theorem is used by:  gruxp  10885  grusucd  45213  grumnudlem  45254