Theorem r1tr2

Description: Each stage of the cumulative hierarchy of sets includes its union, that is, is transitive. JFM CLASSES1 th. 40. (Contributed by FL, 20-Apr-2011)

	Ref	Expression
Assertion	r1tr2	⊢ ∪ ( 𝑅1 ‘ 𝐴 ) ⊆ ( 𝑅1 ‘ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		r1tr	⊢ Tr ( 𝑅1 ‘ 𝐴 )
2		df-tr	⊢ ( Tr ( 𝑅1 ‘ 𝐴 ) ↔ ∪ ( 𝑅1 ‘ 𝐴 ) ⊆ ( 𝑅1 ‘ 𝐴 ) )
3	1 2	mpbi	⊢ ∪ ( 𝑅1 ‘ 𝐴 ) ⊆ ( 𝑅1 ‘ 𝐴 )