Theorem difcom

Description: Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007)

	Ref	Expression
Assertion	difcom	⊢ ( ( 𝐴 ∖ 𝐵 ) ⊆ 𝐶 ↔ ( 𝐴 ∖ 𝐶 ) ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		uncom	⊢ ( 𝐵 ∪ 𝐶 ) = ( 𝐶 ∪ 𝐵 )
2	1	sseq2i	⊢ ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) ↔ 𝐴 ⊆ ( 𝐶 ∪ 𝐵 ) )
3		ssundif	⊢ ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝐴 ∖ 𝐵 ) ⊆ 𝐶 )
4		ssundif	⊢ ( 𝐴 ⊆ ( 𝐶 ∪ 𝐵 ) ↔ ( 𝐴 ∖ 𝐶 ) ⊆ 𝐵 )
5	2 3 4	3bitr3i	⊢ ( ( 𝐴 ∖ 𝐵 ) ⊆ 𝐶 ↔ ( 𝐴 ∖ 𝐶 ) ⊆ 𝐵 )