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 ) ∧ 𝐵 ⊆ 𝐴 ) → 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 ) ) ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) )
3	1 2	sylan	⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) )
4		r1elwf	⊢ ( 𝐵 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) → 𝐵 ∈ ∪ ( 𝑅1 “ On ) )
5	3 4	syl	⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ∈ ∪ ( 𝑅1 “ On ) )