Theorem 2wspdisj

Description: All simple paths of length 2 from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 4-Mar-2018) (Revised by AV, 9-Jan-2022)

	Ref	Expression
Assertion	2wspdisj	⊢ Disj 𝑏 ∈ ( 𝑉 ∖ { 𝐴 } ) ( 𝐴 ( 2 WSPathsNOn 𝐺 ) 𝑏 )

Proof

Step	Hyp	Ref	Expression
1		oveq2	⊢ ( 𝑏 = 𝑐 → ( 𝐴 ( 2 WSPathsNOn 𝐺 ) 𝑏 ) = ( 𝐴 ( 2 WSPathsNOn 𝐺 ) 𝑐 ) )
2		wspthneq1eq2	⊢ ( ( 𝑡 ∈ ( 𝐴 ( 2 WSPathsNOn 𝐺 ) 𝑏 ) ∧ 𝑡 ∈ ( 𝐴 ( 2 WSPathsNOn 𝐺 ) 𝑐 ) ) → ( 𝐴 = 𝐴 ∧ 𝑏 = 𝑐 ) )
3	2	simprd	⊢ ( ( 𝑡 ∈ ( 𝐴 ( 2 WSPathsNOn 𝐺 ) 𝑏 ) ∧ 𝑡 ∈ ( 𝐴 ( 2 WSPathsNOn 𝐺 ) 𝑐 ) ) → 𝑏 = 𝑐 )
4	3	3adant1	⊢ ( ( ⊤ ∧ 𝑡 ∈ ( 𝐴 ( 2 WSPathsNOn 𝐺 ) 𝑏 ) ∧ 𝑡 ∈ ( 𝐴 ( 2 WSPathsNOn 𝐺 ) 𝑐 ) ) → 𝑏 = 𝑐 )
5	1 4	disjord	⊢ ( ⊤ → Disj 𝑏 ∈ ( 𝑉 ∖ { 𝐴 } ) ( 𝐴 ( 2 WSPathsNOn 𝐺 ) 𝑏 ) )
6	5	mptru	⊢ Disj 𝑏 ∈ ( 𝑉 ∖ { 𝐴 } ) ( 𝐴 ( 2 WSPathsNOn 𝐺 ) 𝑏 )