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Theorem wuntr 10105
 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 10104 . . 3 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
21ibi 269 . 2 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
32simp1d 1138 1 (𝑈 ∈ WUni → Tr 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1083   ∈ wcel 2114   ≠ wne 3006  ∀wral 3125  ∅c0 4269  𝒫 cpw 4515  {cpr 4545  ∪ cuni 4814  Tr wtr 5148  WUnicwun 10100 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-v 3475  df-in 3920  df-ss 3930  df-uni 4815  df-tr 5149  df-wun 10102 This theorem is referenced by:  wunelss  10108  intwun  10135
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