Description: Obsolete version of ssct as of 7-Dec-2024. (Contributed by Thierry Arnoux, 31-Jan-2017) (Proof modification is discouraged.) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | ssctOLD | ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω ) → 𝐴 ≼ ω ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ctex | ⊢ ( 𝐵 ≼ ω → 𝐵 ∈ V ) | |
2 | ssdomg | ⊢ ( 𝐵 ∈ V → ( 𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵 ) ) | |
3 | 1 2 | syl | ⊢ ( 𝐵 ≼ ω → ( 𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵 ) ) |
4 | 3 | impcom | ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω ) → 𝐴 ≼ 𝐵 ) |
5 | domtr | ⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ ω ) → 𝐴 ≼ ω ) | |
6 | 4 5 | sylancom | ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω ) → 𝐴 ≼ ω ) |