Description: Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020)
Ref | Expression | ||
---|---|---|---|
Assertion | inabs3 | ⊢ ( 𝐶 ⊆ 𝐵 → ( ( 𝐴 ∩ 𝐵 ) ∩ 𝐶 ) = ( 𝐴 ∩ 𝐶 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inass | ⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ 𝐶 ) = ( 𝐴 ∩ ( 𝐵 ∩ 𝐶 ) ) | |
2 | sseqin2 | ⊢ ( 𝐶 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐶 ) = 𝐶 ) | |
3 | 2 | biimpi | ⊢ ( 𝐶 ⊆ 𝐵 → ( 𝐵 ∩ 𝐶 ) = 𝐶 ) |
4 | 3 | ineq2d | ⊢ ( 𝐶 ⊆ 𝐵 → ( 𝐴 ∩ ( 𝐵 ∩ 𝐶 ) ) = ( 𝐴 ∩ 𝐶 ) ) |
5 | 1 4 | syl5eq | ⊢ ( 𝐶 ⊆ 𝐵 → ( ( 𝐴 ∩ 𝐵 ) ∩ 𝐶 ) = ( 𝐴 ∩ 𝐶 ) ) |