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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 9503 | . . . . . 6 ⊢ 1 ∈ (ℤ≥‘0) | |
2 | eluzfz1 9956 | . . . . . 6 ⊢ (1 ∈ (ℤ≥‘0) → 0 ∈ (0...1)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 0 ∈ (0...1) |
4 | fzsplit 9976 | . . . . 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 2231 | . . 3 ⊢ (𝐴 ∈ (0...1) ↔ 𝐴 ∈ ((0...0) ∪ ((0 + 1)...1))) |
7 | elun 3258 | . . 3 ⊢ (𝐴 ∈ ((0...0) ∪ ((0 + 1)...1)) ↔ (𝐴 ∈ (0...0) ∨ 𝐴 ∈ ((0 + 1)...1))) | |
8 | 6, 7 | bitri 183 | . 2 ⊢ (𝐴 ∈ (0...1) ↔ (𝐴 ∈ (0...0) ∨ 𝐴 ∈ ((0 + 1)...1))) |
9 | elfz1eq 9960 | . . . 4 ⊢ (𝐴 ∈ (0...0) → 𝐴 = 0) | |
10 | 0nn0 9120 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
11 | nn0uz 9491 | . . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0) | |
12 | 10, 11 | eleqtri 2239 | . . . . . 6 ⊢ 0 ∈ (ℤ≥‘0) |
13 | eluzfz1 9956 | . . . . . 6 ⊢ (0 ∈ (ℤ≥‘0) → 0 ∈ (0...0)) | |
14 | 12, 13 | ax-mp 5 | . . . . 5 ⊢ 0 ∈ (0...0) |
15 | eleq1 2227 | . . . . 5 ⊢ (𝐴 = 0 → (𝐴 ∈ (0...0) ↔ 0 ∈ (0...0))) | |
16 | 14, 15 | mpbiri 167 | . . . 4 ⊢ (𝐴 = 0 → 𝐴 ∈ (0...0)) |
17 | 9, 16 | impbii 125 | . . 3 ⊢ (𝐴 ∈ (0...0) ↔ 𝐴 = 0) |
18 | 0p1e1 8962 | . . . . . 6 ⊢ (0 + 1) = 1 | |
19 | 18 | oveq1i 5846 | . . . . 5 ⊢ ((0 + 1)...1) = (1...1) |
20 | 19 | eleq2i 2231 | . . . 4 ⊢ (𝐴 ∈ ((0 + 1)...1) ↔ 𝐴 ∈ (1...1)) |
21 | elfz1eq 9960 | . . . . 5 ⊢ (𝐴 ∈ (1...1) → 𝐴 = 1) | |
22 | 1nn 8859 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
23 | nnuz 9492 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
24 | 22, 23 | eleqtri 2239 | . . . . . . 7 ⊢ 1 ∈ (ℤ≥‘1) |
25 | eluzfz1 9956 | . . . . . . 7 ⊢ (1 ∈ (ℤ≥‘1) → 1 ∈ (1...1)) | |
26 | 24, 25 | ax-mp 5 | . . . . . 6 ⊢ 1 ∈ (1...1) |
27 | eleq1 2227 | . . . . . 6 ⊢ (𝐴 = 1 → (𝐴 ∈ (1...1) ↔ 1 ∈ (1...1))) | |
28 | 26, 27 | mpbiri 167 | . . . . 5 ⊢ (𝐴 = 1 → 𝐴 ∈ (1...1)) |
29 | 21, 28 | impbii 125 | . . . 4 ⊢ (𝐴 ∈ (1...1) ↔ 𝐴 = 1) |
30 | 20, 29 | bitri 183 | . . 3 ⊢ (𝐴 ∈ ((0 + 1)...1) ↔ 𝐴 = 1) |
31 | 17, 30 | orbi12i 754 | . 2 ⊢ ((𝐴 ∈ (0...0) ∨ 𝐴 ∈ ((0 + 1)...1)) ↔ (𝐴 = 0 ∨ 𝐴 = 1)) |
32 | 8, 31 | bitri 183 | 1 ⊢ (𝐴 ∈ (0...1) ↔ (𝐴 = 0 ∨ 𝐴 = 1)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 698 = wceq 1342 ∈ wcel 2135 ∪ cun 3109 ‘cfv 5182 (class class class)co 5836 0cc0 7744 1c1 7745 + caddc 7747 ℕcn 8848 ℕ0cn0 9105 ℤ≥cuz 9457 ...cfz 9935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 |
This theorem is referenced by: hashfiv01gt1 10684 mod2eq1n2dvds 11801 |
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