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Metamath Proof Explorer

Theorem fzo0to2pr

Description: A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017)

Ref Expression
Assertion fzo0to2pr ( 0 ..^ 2 ) = { 0 , 1 }

Proof

Step Hyp Ref Expression
1 2z ⊢ 2 ∈ ℤ
2 fzoval ⊢ ( 2 ∈ ℤ → ( 0 ..^ 2 ) = ( 0 ... ( 2 − 1 ) ) )
3 1 2 ax-mp ⊢ ( 0 ..^ 2 ) = ( 0 ... ( 2 − 1 ) )
4 2m1e1 ⊢ ( 2 − 1 ) = 1
5 0p1e1 ⊢ ( 0 + 1 ) = 1
6 4 5 eqtr4i ⊢ ( 2 − 1 ) = ( 0 + 1 )
7 6 oveq2i ⊢ ( 0 ... ( 2 − 1 ) ) = ( 0 ... ( 0 + 1 ) )
8 0z ⊢ 0 ∈ ℤ
9 fzpr ⊢ ( 0 ∈ ℤ → ( 0 ... ( 0 + 1 ) ) = { 0 , ( 0 + 1 ) } )
10 5 preq2i ⊢ { 0 , ( 0 + 1 ) } = { 0 , 1 }
11 9 10 eqtrdi ⊢ ( 0 ∈ ℤ → ( 0 ... ( 0 + 1 ) ) = { 0 , 1 } )
12 8 11 ax-mp ⊢ ( 0 ... ( 0 + 1 ) ) = { 0 , 1 }
13 3 7 12 3eqtri ⊢ ( 0 ..^ 2 ) = { 0 , 1 }