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Metamath Proof Explorer

Theorem rankid

Description: Identity law for the rank function. (Contributed by NM, 3-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

Ref Expression
Hypothesis rankid.1 ⊢ 𝐴 ∈ V
Assertion rankid 𝐴 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) )

Proof

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