Theorem 0disj

Description: Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016)

	Ref	Expression
Assertion	0disj	⊢ Disj 𝑥 ∈ 𝐴 ∅

Proof

Step	Hyp	Ref	Expression
1		0ss	⊢ ∅ ⊆ { 𝑥 }
2	1	rgenw	⊢ ∀ 𝑥 ∈ 𝐴 ∅ ⊆ { 𝑥 }
3		sndisj	⊢ Disj 𝑥 ∈ 𝐴 { 𝑥 }
4		disjss2	⊢ ( ∀ 𝑥 ∈ 𝐴 ∅ ⊆ { 𝑥 } → ( Disj 𝑥 ∈ 𝐴 { 𝑥 } → Disj 𝑥 ∈ 𝐴 ∅ ) )
5	2 3 4	mp2	⊢ Disj 𝑥 ∈ 𝐴 ∅