MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gruun Structured version   Visualization version   GIF version

Theorem gruun 10550
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 4857 . . . 4 ((𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 1129 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
3 uniiun 4988 . . 3 {𝐴, 𝐵} = 𝑥 ∈ {𝐴, 𝐵}𝑥
42, 3eqtr3di 2793 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → (𝐴𝐵) = 𝑥 ∈ {𝐴, 𝐵}𝑥)
5 simp1 1135 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → 𝑈 ∈ Univ)
6 grupr 10541 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} ∈ 𝑈)
7 vex 3434 . . . . . . 7 𝑥 ∈ V
87elpr 4585 . . . . . 6 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
9 eleq1a 2834 . . . . . . 7 (𝐴𝑈 → (𝑥 = 𝐴𝑥𝑈))
10 eleq1a 2834 . . . . . . 7 (𝐵𝑈 → (𝑥 = 𝐵𝑥𝑈))
119, 10jaao 952 . . . . . 6 ((𝐴𝑈𝐵𝑈) → ((𝑥 = 𝐴𝑥 = 𝐵) → 𝑥𝑈))
128, 11syl5bi 241 . . . . 5 ((𝐴𝑈𝐵𝑈) → (𝑥 ∈ {𝐴, 𝐵} → 𝑥𝑈))
1312ralrimiv 3112 . . . 4 ((𝐴𝑈𝐵𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
14133adant1 1129 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈𝐵𝑈) → ∀𝑥 ∈ {𝐴, 𝐵}𝑥𝑈)
15 gruiun 10543 . . 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 396  wo 844  w3a 1086   = wceq 1539  wcel 2106  wral 3064  cun 3885  {cpr 4564   cuni 4840   ciun 4925  Univcgru 10534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-fv 6435  df-ov 7271  df-oprab 7272  df-mpo 7273  df-map 8605  df-gru 10535
This theorem is referenced by:  gruxp  10551  grusucd  41807  grumnudlem  41862
  Copyright terms: Public domain W3C validator