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Metamath Proof Explorer

Theorem rankel

Description: The membership relation is inherited by the rank function. Proposition 9.16 of TakeutiZaring p. 79. (Contributed by NM, 4-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

Ref Expression
Hypothesis rankel.1 ⊢ B ∈ V
Assertion rankel ⊢ A ∈ B → rank ⁡ A ∈ rank ⁡ B

Proof

Step Hyp Ref Expression
1 rankel.1 ⊢ B ∈ V
2 unir1 ⊢ ⋃ R1 On = V
3 1 2 eleqtrri ⊢ B ∈ ⋃ R1 On
4 rankelb ⊢ B ∈ ⋃ R1 On → A ∈ B → rank ⁡ A ∈ rank ⁡ B
5 3 4 ax-mp ⊢ A ∈ B → rank ⁡ A ∈ rank ⁡ B