Theorem elfzp1

Description: Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005) (Proof shortened by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	elfzp1	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝐾 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ↔ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ∨ 𝐾 = ( 𝑁 + 1 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		fzsuc	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑀 ... ( 𝑁 + 1 ) ) = ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) )
2	1	eleq2d	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝐾 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ↔ 𝐾 ∈ ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ) )
3		elun	⊢ ( 𝐾 ∈ ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ↔ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ∨ 𝐾 ∈ { ( 𝑁 + 1 ) } ) )
4		ovex	⊢ ( 𝑁 + 1 ) ∈ V
5	4	elsn2	⊢ ( 𝐾 ∈ { ( 𝑁 + 1 ) } ↔ 𝐾 = ( 𝑁 + 1 ) )
6	5	orbi2i	⊢ ( ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ∨ 𝐾 ∈ { ( 𝑁 + 1 ) } ) ↔ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ∨ 𝐾 = ( 𝑁 + 1 ) ) )
7	3 6	bitri	⊢ ( 𝐾 ∈ ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) ↔ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ∨ 𝐾 = ( 𝑁 + 1 ) ) )
8	2 7	bitrdi	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝐾 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ↔ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ∨ 𝐾 = ( 𝑁 + 1 ) ) ) )