Theorem fzo0to42pr

Description: A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017)

	Ref	Expression
Assertion	fzo0to42pr	|- ( 0 ..^ 4 ) = ( { 0 , 1 } u. { 2 , 3 } )

Proof

Step	Hyp	Ref	Expression
1		2nn0	|- 2 e. NN0
2		4nn0	|- 4 e. NN0
3		2re	|- 2 e. RR
4		4re	|- 4 e. RR
5		2lt4	|- 2 < 4
6	3 4 5	ltleii	|- 2 <_ 4
7		elfz2nn0	|- ( 2 e. ( 0 ... 4 ) <-> ( 2 e. NN0 /\ 4 e. NN0 /\ 2 <_ 4 ) )
8	1 2 6 7	mpbir3an	|- 2 e. ( 0 ... 4 )
9		fzosplit	|- ( 2 e. ( 0 ... 4 ) -> ( 0 ..^ 4 ) = ( ( 0 ..^ 2 ) u. ( 2 ..^ 4 ) ) )
10	8 9	ax-mp	|- ( 0 ..^ 4 ) = ( ( 0 ..^ 2 ) u. ( 2 ..^ 4 ) )
11		fzo0to2pr	|- ( 0 ..^ 2 ) = { 0 , 1 }
12		4z	|- 4 e. ZZ
13		fzoval	|- ( 4 e. ZZ -> ( 2 ..^ 4 ) = ( 2 ... ( 4 - 1 ) ) )
14	12 13	ax-mp	|- ( 2 ..^ 4 ) = ( 2 ... ( 4 - 1 ) )
15		4m1e3	|- ( 4 - 1 ) = 3
16		df-3	|- 3 = ( 2 + 1 )
17	15 16	eqtri	|- ( 4 - 1 ) = ( 2 + 1 )
18	17	oveq2i	|- ( 2 ... ( 4 - 1 ) ) = ( 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	18 21	eqtri	|- ( 2 ... ( 4 - 1 ) ) = { 2 , ( 2 + 1 ) }
23		2p1e3	|- ( 2 + 1 ) = 3
24	23	preq2i	|- { 2 , ( 2 + 1 ) } = { 2 , 3 }
25	14 22 24	3eqtri	|- ( 2 ..^ 4 ) = { 2 , 3 }
26	11 25	uneq12i	|- ( ( 0 ..^ 2 ) u. ( 2 ..^ 4 ) ) = ( { 0 , 1 } u. { 2 , 3 } )
27	10 26	eqtri	|- ( 0 ..^ 4 ) = ( { 0 , 1 } u. { 2 , 3 } )