Metamath Proof Explorer


Theorem wuntr

Description: A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Assertion wuntr ( 𝑈 ∈ WUni → Tr 𝑈 )

Proof

Step Hyp Ref Expression
1 iswun ( 𝑈 ∈ WUni → ( 𝑈 ∈ WUni ↔ ( Tr 𝑈𝑈 ≠ ∅ ∧ ∀ 𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀ 𝑦𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ) ) ) )
2 1 ibi ( 𝑈 ∈ WUni → ( Tr 𝑈𝑈 ≠ ∅ ∧ ∀ 𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀ 𝑦𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ) ) )
3 2 simp1d ( 𝑈 ∈ WUni → Tr 𝑈 )