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 } )