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Theorem wuntr 10699
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 10698 . . 3 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
21ibi 266 . 2 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
32simp1d 1142 1 (𝑈 ∈ WUni → Tr 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2106  wne 2940  wral 3061  c0 4322  𝒫 cpw 4602  {cpr 4630   cuni 4908  Tr wtr 5265  WUnicwun 10694
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3476  df-in 3955  df-ss 3965  df-uni 4909  df-tr 5266  df-wun 10696
This theorem is referenced by:  wunelss  10702  intwun  10729
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