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

Theorem gruelss 10795
Description: A Grothendieck universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruelss ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)

Proof of Theorem gruelss
StepHypRef Expression
1 grutr 10794 . 2 (𝑈 ∈ Univ → Tr 𝑈)
2 trss 5276 . . 3 (Tr 𝑈 → (𝐴𝑈𝐴𝑈))
32imp 406 . 2 ((Tr 𝑈𝐴𝑈) → 𝐴𝑈)
41, 3sylan 579 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105  wss 3948  Tr wtr 5265  Univcgru 10791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-tr 5266  df-iota 6495  df-fv 6551  df-ov 7415  df-gru 10792
This theorem is referenced by:  gruss  10797  gruuni  10801  gruel  10804  grur1a  10820  grur1  10821
  Copyright terms: Public domain W3C validator