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Metamath Proof Explorer

Theorem rankss

Description: The subset relation is inherited by the rank function. Exercise 1 of TakeutiZaring p. 80. (Contributed by NM, 25-Nov-2003) (Revised by Mario Carneiro, 17-Nov-2014)

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

Proof

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