Theorem difsymssdifssd

Description: If the symmetric difference is contained in C , so is one of the differences. (Contributed by AV, 17-Aug-2022)

	Ref	Expression
Hypothesis	difsymssdifssd.1	⊢ ( 𝜑 → ( 𝐴 △ 𝐵 ) ⊆ 𝐶 )
Assertion	difsymssdifssd	⊢ ( 𝜑 → ( 𝐴 ∖ 𝐵 ) ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		difsymssdifssd.1	⊢ ( 𝜑 → ( 𝐴 △ 𝐵 ) ⊆ 𝐶 )
2		difsssymdif	⊢ ( 𝐴 ∖ 𝐵 ) ⊆ ( 𝐴 △ 𝐵 )
3	2 1	sstrid	⊢ ( 𝜑 → ( 𝐴 ∖ 𝐵 ) ⊆ 𝐶 )