Theorem undisj1

Description: The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004)

	Ref	Expression
Assertion	undisj1	⊢ ( ( ( 𝐴 ∩ 𝐶 ) = ∅ ∧ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ( ( 𝐴 ∪ 𝐵 ) ∩ 𝐶 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		un00	⊢ ( ( ( 𝐴 ∩ 𝐶 ) = ∅ ∧ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐵 ∩ 𝐶 ) ) = ∅ )
2		indir	⊢ ( ( 𝐴 ∪ 𝐵 ) ∩ 𝐶 ) = ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐵 ∩ 𝐶 ) )
3	2	eqeq1i	⊢ ( ( ( 𝐴 ∪ 𝐵 ) ∩ 𝐶 ) = ∅ ↔ ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐵 ∩ 𝐶 ) ) = ∅ )
4	1 3	bitr4i	⊢ ( ( ( 𝐴 ∩ 𝐶 ) = ∅ ∧ ( 𝐵 ∩ 𝐶 ) = ∅ ) ↔ ( ( 𝐴 ∪ 𝐵 ) ∩ 𝐶 ) = ∅ )