This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 ) = ∅ )