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Theorem gruun 10846
Description: A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruun ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) ∈ 𝑈)

Proof of Theorem gruun
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 uniprg 4923 . . . 4 ((𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 1131 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
3 uniiun 5058 . . 3 {𝐴, 𝐵} = 𝑥 ∈ {𝐴, 𝐵}𝑥
42, 3eqtr3di 2792 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) = 𝑥 ∈ {𝐴, 𝐵}𝑥)
5 simp1 1137 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → 𝑈 ∈ Univ)
6 grupr 10837 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
7 vex 3484 . . . . . . 7 𝑥 ∈ V
87elpr 4650 . . . . . 6 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
9 eleq1a 2836 . . . . . . 7 (𝐴𝑈 → (𝑥 = 𝐴𝑥𝑈))
10 eleq1a 2836 . . . . . . 7 (𝐵𝑈 → (𝑥 = 𝐵𝑥𝑈))
119, 10jaao 957 . . . . . 6 ((𝐴𝑈𝐵𝑈) → ((𝑥 = 𝐴𝑥 = 𝐵) → 𝑥𝑈))
128, 11biimtrid 242 . . . . 5 ((𝐴𝑈𝐵𝑈) → (𝑥 ∈ {𝐴, 𝐵} → 𝑥𝑈))
1312ralrimiv 3145 . . . 4 ((𝐴𝑈𝐵𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
14133adant1 1131 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
15 gruiun 10839 . . 3 ((𝑈 ∈ Univ ∧ {𝐴, 𝐵} ∈ 𝑈 ∧ ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈) → 𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
165, 6, 14, 15syl3anc 1373 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → 𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
174, 16eqeltrd 2841 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cun 3949  {cpr 4628   cuni 4907   ciun 4991  Univcgru 10830
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-gru 10831
This theorem is referenced by:  gruxp  10847  grusucd  44249  grumnudlem  44304
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