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Metamath Proof Explorer

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 )