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Theorem gruun 10844
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 4928 . . . 4 ((𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 1129 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
3 uniiun 5063 . . 3 {𝐴, 𝐵} = 𝑥 ∈ {𝐴, 𝐵}𝑥
42, 3eqtr3di 2790 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) = 𝑥 ∈ {𝐴, 𝐵}𝑥)
5 simp1 1135 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → 𝑈 ∈ Univ)
6 grupr 10835 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
7 vex 3482 . . . . . . 7 𝑥 ∈ V
87elpr 4655 . . . . . 6 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
9 eleq1a 2834 . . . . . . 7 (𝐴𝑈 → (𝑥 = 𝐴𝑥𝑈))
10 eleq1a 2834 . . . . . . 7 (𝐵𝑈 → (𝑥 = 𝐵𝑥𝑈))
119, 10jaao 956 . . . . . 6 ((𝐴𝑈𝐵𝑈) → ((𝑥 = 𝐴𝑥 = 𝐵) → 𝑥𝑈))
128, 11biimtrid 242 . . . . 5 ((𝐴𝑈𝐵𝑈) → (𝑥 ∈ {𝐴, 𝐵} → 𝑥𝑈))
1312ralrimiv 3143 . . . 4 ((𝐴𝑈𝐵𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
14133adant1 1129 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
15 gruiun 10837 . . 3 ((𝑈 ∈ Univ ∧ {𝐴, 𝐵} ∈ 𝑈 ∧ ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈) → 𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
165, 6, 14, 15syl3anc 1370 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → 𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
174, 16eqeltrd 2839 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cun 3961  {cpr 4633   cuni 4912   ciun 4996  Univcgru 10828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-gru 10829
This theorem is referenced by:  gruxp  10845  grusucd  44226  grumnudlem  44281
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