Theorem disjss

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 ) )

Proof

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 ) )