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Theorem gruelss 10825
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 10824 . 2 (𝑈 ∈ Univ → Tr 𝑈)
2 trss 5280 . . 3 (Tr 𝑈 → (𝐴𝑈𝐴𝑈))
32imp 405 . 2 ((Tr 𝑈𝐴𝑈) → 𝐴𝑈)
41, 3sylan 578 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  wss 3949  Tr wtr 5269  Univcgru 10821
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 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-tr 5270  df-iota 6505  df-fv 6561  df-ov 7429  df-gru 10822
This theorem is referenced by:  gruss  10827  gruuni  10831  gruel  10834  grur1a  10850  grur1  10851
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