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 } ∪ { 2 , 3 } )

Proof

Step	Hyp	Ref	Expression
1		2nn0	⊢ 2 ∈ ℕ0
2		4nn0	⊢ 4 ∈ ℕ0
3		2re	⊢ 2 ∈ ℝ
4		4re	⊢ 4 ∈ ℝ
5		2lt4	⊢ 2 < 4
6	3 4 5	ltleii	⊢ 2 ≤ 4
7		elfz2nn0	⊢ ( 2 ∈ ( 0 ... 4 ) ↔ ( 2 ∈ ℕ0 ∧ 4 ∈ ℕ0 ∧ 2 ≤ 4 ) )
8	1 2 6 7	mpbir3an	⊢ 2 ∈ ( 0 ... 4 )
9		fzosplit	⊢ ( 2 ∈ ( 0 ... 4 ) → ( 0 ..^ 4 ) = ( ( 0 ..^ 2 ) ∪ ( 2 ..^ 4 ) ) )
10	8 9	ax-mp	⊢ ( 0 ..^ 4 ) = ( ( 0 ..^ 2 ) ∪ ( 2 ..^ 4 ) )
11		fzo0to2pr	⊢ ( 0 ..^ 2 ) = { 0 , 1 }
12		4z	⊢ 4 ∈ ℤ
13		fzoval	⊢ ( 4 ∈ ℤ → ( 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 ∈ ℤ
20		fzpr	⊢ ( 2 ∈ ℤ → ( 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 ) ∪ ( 2 ..^ 4 ) ) = ( { 0 , 1 } ∪ { 2 , 3 } )
27	10 26	eqtri	⊢ ( 0 ..^ 4 ) = ( { 0 , 1 } ∪ { 2 , 3 } )