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Theorem wuntr 10115
Description: A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
wuntr (𝑈 ∈ WUni → Tr 𝑈)

Proof of Theorem wuntr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iswun 10114 . . 3 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
21ibi 268 . 2 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
32simp1d 1134 1 (𝑈 ∈ WUni → Tr 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079  wcel 2105  wne 3013  wral 3135  c0 4288  𝒫 cpw 4535  {cpr 4559   cuni 4830  Tr wtr 5163  WUnicwun 10110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-in 3940  df-ss 3949  df-uni 4831  df-tr 5164  df-wun 10112
This theorem is referenced by:  wunelss  10118  intwun  10145
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