Description: Subclass theorem for disjoints. (Contributed by Peter Mazsa, 28-Oct-2020) (Revised by Peter Mazsa, 22-Sep-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | disjss | ⊢ ( 𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvss | ⊢ ( 𝐴 ⊆ 𝐵 → ◡ 𝐴 ⊆ ◡ 𝐵 ) | |
2 | funALTVss | ⊢ ( ◡ 𝐴 ⊆ ◡ 𝐵 → ( FunALTV ◡ 𝐵 → FunALTV ◡ 𝐴 ) ) | |
3 | 1 2 | syl | ⊢ ( 𝐴 ⊆ 𝐵 → ( FunALTV ◡ 𝐵 → FunALTV ◡ 𝐴 ) ) |
4 | relss | ⊢ ( 𝐴 ⊆ 𝐵 → ( Rel 𝐵 → Rel 𝐴 ) ) | |
5 | 3 4 | anim12d | ⊢ ( 𝐴 ⊆ 𝐵 → ( ( FunALTV ◡ 𝐵 ∧ Rel 𝐵 ) → ( FunALTV ◡ 𝐴 ∧ Rel 𝐴 ) ) ) |
6 | dfdisjALTV | ⊢ ( Disj 𝐵 ↔ ( FunALTV ◡ 𝐵 ∧ Rel 𝐵 ) ) | |
7 | dfdisjALTV | ⊢ ( Disj 𝐴 ↔ ( FunALTV ◡ 𝐴 ∧ Rel 𝐴 ) ) | |
8 | 5 6 7 | 3imtr4g | ⊢ ( 𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴 ) ) |