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Theorem gruelss 10208
 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 10207 . 2 (𝑈 ∈ Univ → Tr 𝑈)
2 trss 5146 . . 3 (Tr 𝑈 → (𝐴𝑈𝐴𝑈))
32imp 410 . 2 ((Tr 𝑈𝐴𝑈) → 𝐴𝑈)
41, 3sylan 583 1 ((𝑈 ∈ Univ ∧ 𝐴𝑈) → 𝐴𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111   ⊆ wss 3881  Tr wtr 5137  Univcgru 10204 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-tr 5138  df-iota 6284  df-fv 6333  df-ov 7139  df-gru 10205 This theorem is referenced by:  gruss  10210  gruuni  10214  gruel  10217  grur1a  10233  grur1  10234
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