Theorem umgr3cyclex

Description: If there are three (different) vertices in a multigraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017) (Revised by AV, 12-Feb-2021)

	Ref	Expression
Hypotheses	uhgr3cyclex.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	uhgr3cyclex.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	umgr3cyclex	⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) ) → ∃ 𝑓 ∃ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ∧ ( ♯ ‘ 𝑓 ) = 3 ∧ ( 𝑝 ‘ 0 ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		uhgr3cyclex.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		uhgr3cyclex.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3		umgruhgr	⊢ ( 𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph )
4	3	3ad2ant1	⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) ) → 𝐺 ∈ UHGraph )
5		simp2	⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) ) → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
6	2	umgredgne	⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐴 , 𝐵 } ∈ 𝐸 ) → 𝐴 ≠ 𝐵 )
7	6	3ad2antr1	⊢ ( ( 𝐺 ∈ UMGraph ∧ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) ) → 𝐴 ≠ 𝐵 )
8		prcom	⊢ { 𝐶 , 𝐴 } = { 𝐴 , 𝐶 }
9	8	eleq1i	⊢ ( { 𝐶 , 𝐴 } ∈ 𝐸 ↔ { 𝐴 , 𝐶 } ∈ 𝐸 )
10	9	biimpi	⊢ ( { 𝐶 , 𝐴 } ∈ 𝐸 → { 𝐴 , 𝐶 } ∈ 𝐸 )
11	10	3ad2ant3	⊢ ( ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) → { 𝐴 , 𝐶 } ∈ 𝐸 )
12	2	umgredgne	⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐴 , 𝐶 } ∈ 𝐸 ) → 𝐴 ≠ 𝐶 )
13	11 12	sylan2	⊢ ( ( 𝐺 ∈ UMGraph ∧ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) ) → 𝐴 ≠ 𝐶 )
14		simp2	⊢ ( ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) → { 𝐵 , 𝐶 } ∈ 𝐸 )
15	2	umgredgne	⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → 𝐵 ≠ 𝐶 )
16	14 15	sylan2	⊢ ( ( 𝐺 ∈ UMGraph ∧ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) ) → 𝐵 ≠ 𝐶 )
17	7 13 16	3jca	⊢ ( ( 𝐺 ∈ UMGraph ∧ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) ) → ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ) )
18	17	3adant2	⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) ) → ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ) )
19		simp3	⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) ) → ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) )
20	1 2	uhgr3cyclex	⊢ ( ( 𝐺 ∈ UHGraph ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ) ) ∧ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) ) → ∃ 𝑓 ∃ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ∧ ( ♯ ‘ 𝑓 ) = 3 ∧ ( 𝑝 ‘ 0 ) = 𝐴 ) )
21	4 5 18 19 20	syl121anc	⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ∧ { 𝐶 , 𝐴 } ∈ 𝐸 ) ) → ∃ 𝑓 ∃ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ∧ ( ♯ ‘ 𝑓 ) = 3 ∧ ( 𝑝 ‘ 0 ) = 𝐴 ) )