Metamath Proof Explorer

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 elin ⊢ ( 𝑥 ∈ ( ( 𝐶 ∖ 𝐵 ) ∩ 𝐴 ) ↔ ( 𝑥 ∈ ( 𝐶 ∖ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) )
6 eldif ⊢ ( 𝑥 ∈ ( 𝐶 ∖ 𝐵 ) ↔ ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵 ) )
7 6 anbi1i ⊢ ( ( 𝑥 ∈ ( 𝐶 ∖ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) ↔ ( ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) )
8 ancom ⊢ ( ( ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵 ) ) )
9 anass ⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶 ) ∧ ¬ 𝑥 ∈ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵 ) ) )
10 8 9 bitr4i ⊢ ( ( ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶 ) ∧ ¬ 𝑥 ∈ 𝐵 ) )
11 5 7 10 3bitri ⊢ ( 𝑥 ∈ ( ( 𝐶 ∖ 𝐵 ) ∩ 𝐴 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶 ) ∧ ¬ 𝑥 ∈ 𝐵 ) )
12 3 4 11 3bitr4g ⊢ ( 𝐴 ⊆ 𝐶 → ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) ↔ 𝑥 ∈ ( ( 𝐶 ∖ 𝐵 ) ∩ 𝐴 ) ) )
13 12 eqrdv ⊢ ( 𝐴 ⊆ 𝐶 → ( 𝐴 ∖ 𝐵 ) = ( ( 𝐶 ∖ 𝐵 ) ∩ 𝐴 ) )