Metamath Proof Explorer


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 ) = ∅ )