Metamath Proof Explorer


Theorem fzo13pr

Description: A 1-based half-open integer interval up to, but not including, 3 is a pair. (Contributed by Thierry Arnoux, 11-Jul-2020)

Ref Expression
Assertion fzo13pr
|- ( 1 ..^ 3 ) = { 1 , 2 }

Proof

Step Hyp Ref Expression
1 3z
 |-  3 e. ZZ
2 fzoval
 |-  ( 3 e. ZZ -> ( 1 ..^ 3 ) = ( 1 ... ( 3 - 1 ) ) )
3 1 2 ax-mp
 |-  ( 1 ..^ 3 ) = ( 1 ... ( 3 - 1 ) )
4 3m1e2
 |-  ( 3 - 1 ) = 2
5 1p1e2
 |-  ( 1 + 1 ) = 2
6 4 5 eqtr4i
 |-  ( 3 - 1 ) = ( 1 + 1 )
7 6 oveq2i
 |-  ( 1 ... ( 3 - 1 ) ) = ( 1 ... ( 1 + 1 ) )
8 1z
 |-  1 e. ZZ
9 fzpr
 |-  ( 1 e. ZZ -> ( 1 ... ( 1 + 1 ) ) = { 1 , ( 1 + 1 ) } )
10 8 9 ax-mp
 |-  ( 1 ... ( 1 + 1 ) ) = { 1 , ( 1 + 1 ) }
11 3 7 10 3eqtri
 |-  ( 1 ..^ 3 ) = { 1 , ( 1 + 1 ) }
12 5 preq2i
 |-  { 1 , ( 1 + 1 ) } = { 1 , 2 }
13 11 12 eqtri
 |-  ( 1 ..^ 3 ) = { 1 , 2 }