Theorem sndisj

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

	Ref	Expression
Assertion	sndisj	⊢ Disj 𝑥 ∈ 𝐴 { 𝑥 }

Proof

Step	Hyp	Ref	Expression
1		dfdisj2	⊢ ( Disj 𝑥 ∈ 𝐴 { 𝑥 } ↔ ∀ 𝑦 ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ { 𝑥 } ) )
2		moeq	⊢ ∃* 𝑥 𝑥 = 𝑦
3		simpr	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ { 𝑥 } ) → 𝑦 ∈ { 𝑥 } )
4		velsn	⊢ ( 𝑦 ∈ { 𝑥 } ↔ 𝑦 = 𝑥 )
5	3 4	sylib	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ { 𝑥 } ) → 𝑦 = 𝑥 )
6	5	equcomd	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ { 𝑥 } ) → 𝑥 = 𝑦 )
7	6	moimi	⊢ ( ∃* 𝑥 𝑥 = 𝑦 → ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ { 𝑥 } ) )
8	2 7	ax-mp	⊢ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ { 𝑥 } )
9	1 8	mpgbir	⊢ Disj 𝑥 ∈ 𝐴 { 𝑥 }