Theorem 2wspiundisj

Description: All simple paths of length 2 from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 5-Mar-2018) (Revised by AV, 14-May-2021) (Proof shortened by AV, 9-Jan-2022)

	Ref	Expression
Assertion	2wspiundisj	⊢ Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ ( 𝑉 ∖ { 𝑎 } ) ( 𝑎 ( 2 WSPathsNOn 𝐺 ) 𝑏 )

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝑎 = 𝑐 → ( 𝑎 ( 2 WSPathsNOn 𝐺 ) 𝑏 ) = ( 𝑐 ( 2 WSPathsNOn 𝐺 ) 𝑏 ) )
2		oveq2	⊢ ( 𝑏 = 𝑑 → ( 𝑐 ( 2 WSPathsNOn 𝐺 ) 𝑏 ) = ( 𝑐 ( 2 WSPathsNOn 𝐺 ) 𝑑 ) )
3		sneq	⊢ ( 𝑎 = 𝑐 → { 𝑎 } = { 𝑐 } )
4	3	difeq2d	⊢ ( 𝑎 = 𝑐 → ( 𝑉 ∖ { 𝑎 } ) = ( 𝑉 ∖ { 𝑐 } ) )
5		wspthneq1eq2	⊢ ( ( 𝑡 ∈ ( 𝑎 ( 2 WSPathsNOn 𝐺 ) 𝑏 ) ∧ 𝑡 ∈ ( 𝑐 ( 2 WSPathsNOn 𝐺 ) 𝑑 ) ) → ( 𝑎 = 𝑐 ∧ 𝑏 = 𝑑 ) )
6	5	simpld	⊢ ( ( 𝑡 ∈ ( 𝑎 ( 2 WSPathsNOn 𝐺 ) 𝑏 ) ∧ 𝑡 ∈ ( 𝑐 ( 2 WSPathsNOn 𝐺 ) 𝑑 ) ) → 𝑎 = 𝑐 )
7	6	3adant1	⊢ ( ( ⊤ ∧ 𝑡 ∈ ( 𝑎 ( 2 WSPathsNOn 𝐺 ) 𝑏 ) ∧ 𝑡 ∈ ( 𝑐 ( 2 WSPathsNOn 𝐺 ) 𝑑 ) ) → 𝑎 = 𝑐 )
8	1 2 4 7	disjiund	⊢ ( ⊤ → Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ ( 𝑉 ∖ { 𝑎 } ) ( 𝑎 ( 2 WSPathsNOn 𝐺 ) 𝑏 ) )
9	8	mptru	⊢ Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ ( 𝑉 ∖ { 𝑎 } ) ( 𝑎 ( 2 WSPathsNOn 𝐺 ) 𝑏 )