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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 8662 | . . . . . 6 ⊢ 1 ∈ (ℤ≥‘0) | |
2 | eluzfz1 9050 | . . . . . 6 ⊢ (1 ∈ (ℤ≥‘0) → 0 ∈ (0...1)) | |
3 | 1, 2 | ax-mp 7 | . . . . 5 ⊢ 0 ∈ (0...1) |
4 | fzsplit 9070 | . . . . 5 ⊢ (0 ∈ (0...1) → (0...1) = ((0...0) ∪ ((0 + 1)...1))) | |
5 | 3, 4 | ax-mp 7 | . . . 4 ⊢ (0...1) = ((0...0) ∪ ((0 + 1)...1)) |
6 | 5 | eleq2i 2145 | . . 3 ⊢ (𝐴 ∈ (0...1) ↔ 𝐴 ∈ ((0...0) ∪ ((0 + 1)...1))) |
7 | elun 3113 | . . 3 ⊢ (𝐴 ∈ ((0...0) ∪ ((0 + 1)...1)) ↔ (𝐴 ∈ (0...0) ∨ 𝐴 ∈ ((0 + 1)...1))) | |
8 | 6, 7 | bitri 182 | . 2 ⊢ (𝐴 ∈ (0...1) ↔ (𝐴 ∈ (0...0) ∨ 𝐴 ∈ ((0 + 1)...1))) |
9 | elfz1eq 9054 | . . . 4 ⊢ (𝐴 ∈ (0...0) → 𝐴 = 0) | |
10 | 0nn0 8303 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
11 | nn0uz 8653 | . . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0) | |
12 | 10, 11 | eleqtri 2153 | . . . . . 6 ⊢ 0 ∈ (ℤ≥‘0) |
13 | eluzfz1 9050 | . . . . . 6 ⊢ (0 ∈ (ℤ≥‘0) → 0 ∈ (0...0)) | |
14 | 12, 13 | ax-mp 7 | . . . . 5 ⊢ 0 ∈ (0...0) |
15 | eleq1 2141 | . . . . 5 ⊢ (𝐴 = 0 → (𝐴 ∈ (0...0) ↔ 0 ∈ (0...0))) | |
16 | 14, 15 | mpbiri 166 | . . . 4 ⊢ (𝐴 = 0 → 𝐴 ∈ (0...0)) |
17 | 9, 16 | impbii 124 | . . 3 ⊢ (𝐴 ∈ (0...0) ↔ 𝐴 = 0) |
18 | 0p1e1 8153 | . . . . . 6 ⊢ (0 + 1) = 1 | |
19 | 18 | oveq1i 5542 | . . . . 5 ⊢ ((0 + 1)...1) = (1...1) |
20 | 19 | eleq2i 2145 | . . . 4 ⊢ (𝐴 ∈ ((0 + 1)...1) ↔ 𝐴 ∈ (1...1)) |
21 | elfz1eq 9054 | . . . . 5 ⊢ (𝐴 ∈ (1...1) → 𝐴 = 1) | |
22 | 1nn 8050 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
23 | nnuz 8654 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
24 | 22, 23 | eleqtri 2153 | . . . . . . 7 ⊢ 1 ∈ (ℤ≥‘1) |
25 | eluzfz1 9050 | . . . . . . 7 ⊢ (1 ∈ (ℤ≥‘1) → 1 ∈ (1...1)) | |
26 | 24, 25 | ax-mp 7 | . . . . . 6 ⊢ 1 ∈ (1...1) |
27 | eleq1 2141 | . . . . . 6 ⊢ (𝐴 = 1 → (𝐴 ∈ (1...1) ↔ 1 ∈ (1...1))) | |
28 | 26, 27 | mpbiri 166 | . . . . 5 ⊢ (𝐴 = 1 → 𝐴 ∈ (1...1)) |
29 | 21, 28 | impbii 124 | . . . 4 ⊢ (𝐴 ∈ (1...1) ↔ 𝐴 = 1) |
30 | 20, 29 | bitri 182 | . . 3 ⊢ (𝐴 ∈ ((0 + 1)...1) ↔ 𝐴 = 1) |
31 | 17, 30 | orbi12i 713 | . 2 ⊢ ((𝐴 ∈ (0...0) ∨ 𝐴 ∈ ((0 + 1)...1)) ↔ (𝐴 = 0 ∨ 𝐴 = 1)) |
32 | 8, 31 | bitri 182 | 1 ⊢ (𝐴 ∈ (0...1) ↔ (𝐴 = 0 ∨ 𝐴 = 1)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∨ wo 661 = wceq 1284 ∈ wcel 1433 ∪ cun 2971 ‘cfv 4922 (class class class)co 5532 0cc0 6981 1c1 6982 + caddc 6984 ℕcn 8039 ℕ0cn0 8288 ℤ≥cuz 8619 ...cfz 9029 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-fz 9030 |
This theorem is referenced by: mod2eq1n2dvds 10279 |
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