Theorem rankel

Description: The membership relation is inherited by the rank function. Proposition 9.16 of TakeutiZaring p. 79. (Contributed by NM, 4-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	rankel.1	⊢ 𝐵 ∈ V
Assertion	rankel	⊢ ( 𝐴 ∈ 𝐵 → ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		rankel.1	⊢ 𝐵 ∈ V
2		unir1	⊢ ∪ ( 𝑅1 “ On ) = V
3	1 2	eleqtrri	⊢ 𝐵 ∈ ∪ ( 𝑅1 “ On )
4		rankelb	⊢ ( 𝐵 ∈ ∪ ( 𝑅1 “ On ) → ( 𝐴 ∈ 𝐵 → ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐵 ) ) )
5	3 4	ax-mp	⊢ ( 𝐴 ∈ 𝐵 → ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐵 ) )