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Metamath Proof Explorer

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 ) = ∅ )