Theorem rankvaln

Description: Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 , unless A is a proper class.) (Contributed by Mario Carneiro, 22-Mar-2013) (Revised by Mario Carneiro, 10-Sep-2013)

	Ref	Expression
Assertion	rankvaln	⊢ ( ¬ 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( rank ‘ 𝐴 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		rankf	⊢ rank : ∪ ( 𝑅1 “ On ) ⟶ On
2	1	fdmi	⊢ dom rank = ∪ ( 𝑅1 “ On )
3	2	eleq2i	⊢ ( 𝐴 ∈ dom rank ↔ 𝐴 ∈ ∪ ( 𝑅1 “ On ) )
4		ndmfv	⊢ ( ¬ 𝐴 ∈ dom rank → ( rank ‘ 𝐴 ) = ∅ )
5	3 4	sylnbir	⊢ ( ¬ 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( rank ‘ 𝐴 ) = ∅ )