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 } ∪ { 2 , 3 } )

Proof

Step	Hyp	Ref	Expression
1		1nn0	⊢ 1 ∈ ℕ0
2		3nn0	⊢ 3 ∈ ℕ0
3		1le3	⊢ 1 ≤ 3
4		elfz2nn0	⊢ ( 1 ∈ ( 0 ... 3 ) ↔ ( 1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3 ) )
5	1 2 3 4	mpbir3an	⊢ 1 ∈ ( 0 ... 3 )
6		fzsplit	⊢ ( 1 ∈ ( 0 ... 3 ) → ( 0 ... 3 ) = ( ( 0 ... 1 ) ∪ ( ( 1 + 1 ) ... 3 ) ) )
7	5 6	ax-mp	⊢ ( 0 ... 3 ) = ( ( 0 ... 1 ) ∪ ( ( 1 + 1 ) ... 3 ) )
8		1e0p1	⊢ 1 = ( 0 + 1 )
9	8	oveq2i	⊢ ( 0 ... 1 ) = ( 0 ... ( 0 + 1 ) )
10		0z	⊢ 0 ∈ ℤ
11		fzpr	⊢ ( 0 ∈ ℤ → ( 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 ∈ ℤ
20		fzpr	⊢ ( 2 ∈ ℤ → ( 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 ) ∪ ( ( 1 + 1 ) ... 3 ) ) = ( { 0 , 1 } ∪ { 2 , 3 } )
26	7 25	eqtri	⊢ ( 0 ... 3 ) = ( { 0 , 1 } ∪ { 2 , 3 } )