Metamath Proof Explorer


Theorem sswf

Description: A subset of a well-founded set is well-founded. (Contributed by Mario Carneiro, 17-Nov-2014)

Ref Expression
Assertion sswf ( ( 𝐴 ( 𝑅1 “ On ) ∧ 𝐵𝐴 ) → 𝐵 ( 𝑅1 “ On ) )

Proof

Step Hyp Ref Expression
1 rankidb ( 𝐴 ( 𝑅1 “ On ) → 𝐴 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) )
2 r1sscl ( ( 𝐴 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) ∧ 𝐵𝐴 ) → 𝐵 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) )
3 1 2 sylan ( ( 𝐴 ( 𝑅1 “ On ) ∧ 𝐵𝐴 ) → 𝐵 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) )
4 r1elwf ( 𝐵 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) → 𝐵 ( 𝑅1 “ On ) )
5 3 4 syl ( ( 𝐴 ( 𝑅1 “ On ) ∧ 𝐵𝐴 ) → 𝐵 ( 𝑅1 “ On ) )