Theorem grutr

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

	Ref	Expression
Assertion	grutr	⊢ ( 𝑈 ∈ Univ → Tr 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		elgrug	⊢ ( 𝑈 ∈ Univ → ( 𝑈 ∈ Univ ↔ ( Tr 𝑈 ∧ ∀ 𝑥 ∈ 𝑈 ( 𝒫 𝑥 ∈ 𝑈 ∧ ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ∧ ∀ 𝑦 ∈ ( 𝑈 ↑m 𝑥 ) ∪ ran 𝑦 ∈ 𝑈 ) ) ) )
2	1	ibi	⊢ ( 𝑈 ∈ Univ → ( Tr 𝑈 ∧ ∀ 𝑥 ∈ 𝑈 ( 𝒫 𝑥 ∈ 𝑈 ∧ ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ∧ ∀ 𝑦 ∈ ( 𝑈 ↑m 𝑥 ) ∪ ran 𝑦 ∈ 𝑈 ) ) )
3	2	simpld	⊢ ( 𝑈 ∈ Univ → Tr 𝑈 )