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Theorem grutr 10012
Description: A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
grutr (𝑈 ∈ Univ → Tr 𝑈)

Proof of Theorem grutr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elgrug 10011 . . 3 (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈))))
21ibi 259 . 2 (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈𝑚 𝑥) ran 𝑦𝑈)))
32simpld 487 1 (𝑈 ∈ Univ → Tr 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1069  wcel 2051  wral 3083  𝒫 cpw 4417  {cpr 4438   cuni 4709  Tr wtr 5027  ran crn 5405  (class class class)co 6975  𝑚 cmap 8205  Univcgru 10009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-tr 5028  df-iota 6150  df-fv 6194  df-ov 6978  df-gru 10010
This theorem is referenced by:  gruelss  10013  gruwun  10032  intgru  10033  gruina  10037  grur1  10039  grutsk  10041
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