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Metamath Proof Explorer

Theorem usgr2trlncrct

Description: In a simple graph, any trail of length 2 is not a circuit. (Contributed by AV, 5-Jun-2021)

Ref Expression
Assertion usgr2trlncrct ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ) → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → ¬ 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 ) )

Proof

Step Hyp Ref Expression
1 usgr2trlncl ⊢ ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ) → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) )
2 1 imp ⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ) ∧ 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ) → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) )
3 crctprop ⊢ ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) )
4 fveq2 ⊢ ( ( ♯ ‘ 𝐹 ) = 2 → ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = ( 𝑃 ‘ 2 ) )
5 4 eqeq2d ⊢ ( ( ♯ ‘ 𝐹 ) = 2 → ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ↔ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 2 ) ) )
6 5 biimpcd ⊢ ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) → ( ( ♯ ‘ 𝐹 ) = 2 → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 2 ) ) )
7 3 6 simpl2im ⊢ ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 → ( ( ♯ ‘ 𝐹 ) = 2 → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 2 ) ) )
8 7 com12 ⊢ ( ( ♯ ‘ 𝐹 ) = 2 → ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 2 ) ) )
9 8 ad2antlr ⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ) ∧ 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ) → ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 2 ) ) )
10 9 necon3ad ⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ) ∧ 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ) → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) → ¬ 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 ) )
11 2 10 mpd ⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ) ∧ 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ) → ¬ 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 )
12 11 ex ⊢ ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ) → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → ¬ 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 ) )