Theorem wuntr 10392

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.

Step	Hyp	Ref	Expression
1		iswun 10391	. . 3 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))
2	1	ibi 266	. 2 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))
3	2	simp1d 1140	1 ⊢ (𝑈 ∈ WUni → Tr 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1085   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∅c0 4253  𝒫 cpw 4530  {cpr 4560  ∪ cuni 4836  Tr wtr 5187  WUnicwun 10387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-v 3424  df-in 3890  df-ss 3900  df-uni 4837  df-tr 5188  df-wun 10389

This theorem is referenced by:  wunelss  10395  intwun  10422