Theorem gruelss

Description: A Grothendieck universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013)

	Ref	Expression
Assertion	gruelss	⊢ ( ( 𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ) → 𝐴 ⊆ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		grutr	⊢ ( 𝑈 ∈ Univ → Tr 𝑈 )
2		trss	⊢ ( Tr 𝑈 → ( 𝐴 ∈ 𝑈 → 𝐴 ⊆ 𝑈 ) )
3	2	imp	⊢ ( ( Tr 𝑈 ∧ 𝐴 ∈ 𝑈 ) → 𝐴 ⊆ 𝑈 )
4	1 3	sylan	⊢ ( ( 𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ) → 𝐴 ⊆ 𝑈 )