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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 } ∪ { 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 } )