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Metamath Proof Explorer

Theorem fz0to3un2pr

Description: An integer range from 0 to 3 is the union of two unordered pairs. (Contributed by AV, 7-Feb-2021)

Ref Expression
Assertion fz0to3un2pr 
|- ( 0 ... 3 ) = ( { 0 , 1 } u. { 2 , 3 } )

Proof

Step Hyp Ref Expression
1 1nn0 
 |-  1 e. NN0
2 3nn0 
 |-  3 e. NN0
3 1le3 
 |-  1 <_ 3
4 elfz2nn0 
 |-  ( 1 e. ( 0 ... 3 ) <-> ( 1 e. NN0 /\ 3 e. NN0 /\ 1 <_ 3 ) )
5 1 2 3 4 mpbir3an 
 |-  1 e. ( 0 ... 3 )
6 fzsplit 
 |-  ( 1 e. ( 0 ... 3 ) -> ( 0 ... 3 ) = ( ( 0 ... 1 ) u. ( ( 1 + 1 ) ... 3 ) ) )
7 5 6 ax-mp 
 |-  ( 0 ... 3 ) = ( ( 0 ... 1 ) u. ( ( 1 + 1 ) ... 3 ) )
8 1e0p1 
 |-  1 = ( 0 + 1 )
9 8 oveq2i 
 |-  ( 0 ... 1 ) = ( 0 ... ( 0 + 1 ) )
10 0z 
 |-  0 e. ZZ
11 fzpr 
 |-  ( 0 e. ZZ -> ( 0 ... ( 0 + 1 ) ) = { 0 , ( 0 + 1 ) } )
12 10 11 ax-mp 
 |-  ( 0 ... ( 0 + 1 ) ) = { 0 , ( 0 + 1 ) }
13 0p1e1 
 |-  ( 0 + 1 ) = 1
14 13 preq2i 
 |-  { 0 , ( 0 + 1 ) } = { 0 , 1 }
15 9 12 14 3eqtri 
 |-  ( 0 ... 1 ) = { 0 , 1 }
16 1p1e2 
 |-  ( 1 + 1 ) = 2
17 df-3 
 |-  3 = ( 2 + 1 )
18 16 17 oveq12i 
 |-  ( ( 1 + 1 ) ... 3 ) = ( 2 ... ( 2 + 1 ) )
19 2z 
 |-  2 e. ZZ
20 fzpr 
 |-  ( 2 e. ZZ -> ( 2 ... ( 2 + 1 ) ) = { 2 , ( 2 + 1 ) } )
21 19 20 ax-mp 
 |-  ( 2 ... ( 2 + 1 ) ) = { 2 , ( 2 + 1 ) }
22 2p1e3 
 |-  ( 2 + 1 ) = 3
23 22 preq2i 
 |-  { 2 , ( 2 + 1 ) } = { 2 , 3 }
24 18 21 23 3eqtri 
 |-  ( ( 1 + 1 ) ... 3 ) = { 2 , 3 }
25 15 24 uneq12i 
 |-  ( ( 0 ... 1 ) u. ( ( 1 + 1 ) ... 3 ) ) = ( { 0 , 1 } u. { 2 , 3 } )
26 7 25 eqtri 
 |-  ( 0 ... 3 ) = ( { 0 , 1 } u. { 2 , 3 } )