Theorem sscon

Description: Contraposition law for subsets. Exercise 15 of TakeutiZaring p. 22. (Contributed by NM, 22-Mar-1998)

	Ref	Expression
Assertion	sscon	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∖ 𝐵 ) ⊆ ( 𝐶 ∖ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ssel	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
2	1	con3d	⊢ ( 𝐴 ⊆ 𝐵 → ( ¬ 𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴 ) )
3	2	anim2d	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵 ) → ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴 ) ) )
4		eldif	⊢ ( 𝑥 ∈ ( 𝐶 ∖ 𝐵 ) ↔ ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵 ) )
5		eldif	⊢ ( 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ↔ ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴 ) )
6	3 4 5	3imtr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ ( 𝐶 ∖ 𝐵 ) → 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ) )
7	6	ssrdv	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∖ 𝐵 ) ⊆ ( 𝐶 ∖ 𝐴 ) )