Theorem rankeq0

Description: A set is empty iff its rank is empty. (Contributed by NM, 18-Sep-2006) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	rankeq0.1	⊢ 𝐴 ∈ V
Assertion	rankeq0	⊢ ( 𝐴 = ∅ ↔ ( rank ‘ 𝐴 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		rankeq0.1	⊢ 𝐴 ∈ V
2		unir1	⊢ ∪ ( 𝑅1 “ On ) = V
3	1 2	eleqtrri	⊢ 𝐴 ∈ ∪ ( 𝑅1 “ On )
4		rankeq0b	⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( 𝐴 = ∅ ↔ ( rank ‘ 𝐴 ) = ∅ ) )
5	3 4	ax-mp	⊢ ( 𝐴 = ∅ ↔ ( rank ‘ 𝐴 ) = ∅ )