Theorem difin2

Description: Represent a class difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009)

	Ref	Expression
Assertion	difin2	⊢ ( 𝐴 ⊆ 𝐶 → ( 𝐴 ∖ 𝐵 ) = ( ( 𝐶 ∖ 𝐵 ) ∩ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ssel	⊢ ( 𝐴 ⊆ 𝐶 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶 ) )
2	1	pm4.71d	⊢ ( 𝐴 ⊆ 𝐶 → ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶 ) ) )
3	2	anbi1d	⊢ ( 𝐴 ⊆ 𝐶 → ( ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶 ) ∧ ¬ 𝑥 ∈ 𝐵 ) ) )
4		eldif	⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) )
5		ancom	⊢ ( ( ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵 ) ) )
6		elin	⊢ ( 𝑥 ∈ ( ( 𝐶 ∖ 𝐵 ) ∩ 𝐴 ) ↔ ( 𝑥 ∈ ( 𝐶 ∖ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) )
7		eldif	⊢ ( 𝑥 ∈ ( 𝐶 ∖ 𝐵 ) ↔ ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵 ) )
8	6 7	bianbi	⊢ ( 𝑥 ∈ ( ( 𝐶 ∖ 𝐵 ) ∩ 𝐴 ) ↔ ( ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) )
9		anass	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶 ) ∧ ¬ 𝑥 ∈ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵 ) ) )
10	5 8 9	3bitr4i	⊢ ( 𝑥 ∈ ( ( 𝐶 ∖ 𝐵 ) ∩ 𝐴 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶 ) ∧ ¬ 𝑥 ∈ 𝐵 ) )
11	3 4 10	3bitr4g	⊢ ( 𝐴 ⊆ 𝐶 → ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) ↔ 𝑥 ∈ ( ( 𝐶 ∖ 𝐵 ) ∩ 𝐴 ) ) )
12	11	eqrdv	⊢ ( 𝐴 ⊆ 𝐶 → ( 𝐴 ∖ 𝐵 ) = ( ( 𝐶 ∖ 𝐵 ) ∩ 𝐴 ) )