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Mirrors > Home > ILE Home > Th. List > fz01or | GIF version |
Description: An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.) |
Ref | Expression |
---|---|
fz01or | ⊢ (𝐴 ∈ (0...1) ↔ (𝐴 = 0 ∨ 𝐴 = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1eluzge0 9550 | . . . . . 6 ⊢ 1 ∈ (ℤ≥‘0) | |
2 | eluzfz1 10004 | . . . . . 6 ⊢ (1 ∈ (ℤ≥‘0) → 0 ∈ (0...1)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 0 ∈ (0...1) |
4 | fzsplit 10024 | . . . . 5 ⊢ (0 ∈ (0...1) → (0...1) = ((0...0) ∪ ((0 + 1)...1))) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (0...1) = ((0...0) ∪ ((0 + 1)...1)) |
6 | 5 | eleq2i 2244 | . . 3 ⊢ (𝐴 ∈ (0...1) ↔ 𝐴 ∈ ((0...0) ∪ ((0 + 1)...1))) |
7 | elun 3276 | . . 3 ⊢ (𝐴 ∈ ((0...0) ∪ ((0 + 1)...1)) ↔ (𝐴 ∈ (0...0) ∨ 𝐴 ∈ ((0 + 1)...1))) | |
8 | 6, 7 | bitri 184 | . 2 ⊢ (𝐴 ∈ (0...1) ↔ (𝐴 ∈ (0...0) ∨ 𝐴 ∈ ((0 + 1)...1))) |
9 | elfz1eq 10008 | . . . 4 ⊢ (𝐴 ∈ (0...0) → 𝐴 = 0) | |
10 | 0nn0 9167 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
11 | nn0uz 9538 | . . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0) | |
12 | 10, 11 | eleqtri 2252 | . . . . . 6 ⊢ 0 ∈ (ℤ≥‘0) |
13 | eluzfz1 10004 | . . . . . 6 ⊢ (0 ∈ (ℤ≥‘0) → 0 ∈ (0...0)) | |
14 | 12, 13 | ax-mp 5 | . . . . 5 ⊢ 0 ∈ (0...0) |
15 | eleq1 2240 | . . . . 5 ⊢ (𝐴 = 0 → (𝐴 ∈ (0...0) ↔ 0 ∈ (0...0))) | |
16 | 14, 15 | mpbiri 168 | . . . 4 ⊢ (𝐴 = 0 → 𝐴 ∈ (0...0)) |
17 | 9, 16 | impbii 126 | . . 3 ⊢ (𝐴 ∈ (0...0) ↔ 𝐴 = 0) |
18 | 0p1e1 9009 | . . . . . 6 ⊢ (0 + 1) = 1 | |
19 | 18 | oveq1i 5878 | . . . . 5 ⊢ ((0 + 1)...1) = (1...1) |
20 | 19 | eleq2i 2244 | . . . 4 ⊢ (𝐴 ∈ ((0 + 1)...1) ↔ 𝐴 ∈ (1...1)) |
21 | elfz1eq 10008 | . . . . 5 ⊢ (𝐴 ∈ (1...1) → 𝐴 = 1) | |
22 | 1nn 8906 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
23 | nnuz 9539 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
24 | 22, 23 | eleqtri 2252 | . . . . . . 7 ⊢ 1 ∈ (ℤ≥‘1) |
25 | eluzfz1 10004 | . . . . . . 7 ⊢ (1 ∈ (ℤ≥‘1) → 1 ∈ (1...1)) | |
26 | 24, 25 | ax-mp 5 | . . . . . 6 ⊢ 1 ∈ (1...1) |
27 | eleq1 2240 | . . . . . 6 ⊢ (𝐴 = 1 → (𝐴 ∈ (1...1) ↔ 1 ∈ (1...1))) | |
28 | 26, 27 | mpbiri 168 | . . . . 5 ⊢ (𝐴 = 1 → 𝐴 ∈ (1...1)) |
29 | 21, 28 | impbii 126 | . . . 4 ⊢ (𝐴 ∈ (1...1) ↔ 𝐴 = 1) |
30 | 20, 29 | bitri 184 | . . 3 ⊢ (𝐴 ∈ ((0 + 1)...1) ↔ 𝐴 = 1) |
31 | 17, 30 | orbi12i 764 | . 2 ⊢ ((𝐴 ∈ (0...0) ∨ 𝐴 ∈ ((0 + 1)...1)) ↔ (𝐴 = 0 ∨ 𝐴 = 1)) |
32 | 8, 31 | bitri 184 | 1 ⊢ (𝐴 ∈ (0...1) ↔ (𝐴 = 0 ∨ 𝐴 = 1)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 ∪ cun 3127 ‘cfv 5211 (class class class)co 5868 0cc0 7789 1c1 7790 + caddc 7792 ℕcn 8895 ℕ0cn0 9152 ℤ≥cuz 9504 ...cfz 9982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-inn 8896 df-n0 9153 df-z 9230 df-uz 9505 df-fz 9983 |
This theorem is referenced by: hashfiv01gt1 10733 mod2eq1n2dvds 11854 |
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