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Metamath Proof Explorer

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 ) ) ) )