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

Theorem gruuni 10777
Description: A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.)
Assertion
Ref Expression
gruuni ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)

Proof of Theorem gruuni
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 uniiun 5054 . 2 𝐴 = 𝑥𝐴 𝑥
2 gruelss 10771 . . . 4 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
3 dfss3 3966 . . . 4 (𝐴𝑈 ↔ ∀𝑥𝐴 𝑥𝑈)
42, 3sylib 217 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → ∀𝑥𝐴 𝑥𝑈)
5 gruiun 10776 . . 3 ((𝑈 ∈ Univ ∧ 𝐴𝑈 ∧ ∀𝑥𝐴 𝑥𝑈) → 𝑥𝐴 𝑥𝑈)
64, 5mpd3an3 1462 . 2 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝑥𝐴 𝑥𝑈)
71, 6eqeltrid 2836 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wral 3060  wss 3944   cuni 4901   ciun 4990  Univcgru 10767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-map 8805  df-gru 10768
This theorem is referenced by:  gruwun  10790  gruina  10795  grumnudlem  42813
  Copyright terms: Public domain W3C validator