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Theorem gruelss 10791
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 10790 . 2 (𝑈 ∈ Univ → Tr 𝑈)
2 trss 5269 . . 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 2098  wss 3943  Tr wtr 5258  Univcgru 10787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-tr 5259  df-iota 6489  df-fv 6545  df-ov 7408  df-gru 10788
This theorem is referenced by:  gruss  10793  gruuni  10797  gruel  10800  grur1a  10816  grur1  10817
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