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Theorem gruelss 10832
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 10831 . 2 (𝑈 ∈ Univ → Tr 𝑈)
2 trss 5276 . . 3 (Tr 𝑈 → (𝐴𝑈𝐴𝑈))
32imp 406 . 2 ((Tr 𝑈𝐴𝑈) → 𝐴𝑈)
41, 3sylan 580 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wss 3963  Tr wtr 5265  Univcgru 10828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-tr 5266  df-iota 6516  df-fv 6571  df-ov 7434  df-gru 10829
This theorem is referenced by:  gruss  10834  gruuni  10838  gruel  10841  grur1a  10857  grur1  10858
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