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 ) ) |
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 ) ) |