Theorem clnbupgreli

Description: A member of the closed neighborhood of a vertex in a pseudograph. (Contributed by AV, 28-Dec-2025)

	Ref	Expression
Hypothesis	clnbupgreli.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	clnbupgreli	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) ) → ( 𝑁 = 𝐾 ∨ { 𝑁 , 𝐾 } ∈ 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		clnbupgreli.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
2		simpr	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) ) → 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) )
3		simpl	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) ) → 𝐺 ∈ UPGraph )
4		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
5	4	clnbgrcl	⊢ ( 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) → 𝐾 ∈ ( Vtx ‘ 𝐺 ) )
6	5	adantl	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) ) → 𝐾 ∈ ( Vtx ‘ 𝐺 ) )
7	4	clnbgrisvtx	⊢ ( 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) → 𝑁 ∈ ( Vtx ‘ 𝐺 ) )
8	7	adantl	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) ) → 𝑁 ∈ ( Vtx ‘ 𝐺 ) )
9	4 1	clnbupgrel	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐾 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) ↔ ( 𝑁 = 𝐾 ∨ { 𝑁 , 𝐾 } ∈ 𝐸 ) ) )
10	3 6 8 9	syl3anc	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) ) → ( 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) ↔ ( 𝑁 = 𝐾 ∨ { 𝑁 , 𝐾 } ∈ 𝐸 ) ) )
11	2 10	mpbid	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) ) → ( 𝑁 = 𝐾 ∨ { 𝑁 , 𝐾 } ∈ 𝐸 ) )