Description: A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017)
Ref | Expression | ||
---|---|---|---|
Assertion | wuntr | ⊢ ( 𝑈 ∈ WUni → Tr 𝑈 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iswun | ⊢ ( 𝑈 ∈ WUni → ( 𝑈 ∈ WUni ↔ ( Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝑈 ( ∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ) ) ) ) | |
2 | 1 | ibi | ⊢ ( 𝑈 ∈ WUni → ( Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝑈 ( ∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ) ) ) |
3 | 2 | simp1d | ⊢ ( 𝑈 ∈ WUni → Tr 𝑈 ) |