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Theorem wuntr 10696
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 10695 . . 3 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
21ibi 267 . 2 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
32simp1d 1139 1 (𝑈 ∈ WUni → Tr 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2098  wne 2932  wral 3053  c0 4314  𝒫 cpw 4594  {cpr 4622   cuni 4899  Tr wtr 5255  WUnicwun 10691
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 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-v 3468  df-in 3947  df-ss 3957  df-uni 4900  df-tr 5256  df-wun 10693
This theorem is referenced by:  wunelss  10699  intwun  10726
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