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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 9362 | . . . . . 6 ⊢ 1 ∈ (ℤ≥‘0) | |
2 | eluzfz1 9804 | . . . . . 6 ⊢ (1 ∈ (ℤ≥‘0) → 0 ∈ (0...1)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 0 ∈ (0...1) |
4 | fzsplit 9824 | . . . . 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 2204 | . . 3 ⊢ (𝐴 ∈ (0...1) ↔ 𝐴 ∈ ((0...0) ∪ ((0 + 1)...1))) |
7 | elun 3212 | . . 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 9808 | . . . 4 ⊢ (𝐴 ∈ (0...0) → 𝐴 = 0) | |
10 | 0nn0 8985 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
11 | nn0uz 9353 | . . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0) | |
12 | 10, 11 | eleqtri 2212 | . . . . . 6 ⊢ 0 ∈ (ℤ≥‘0) |
13 | eluzfz1 9804 | . . . . . 6 ⊢ (0 ∈ (ℤ≥‘0) → 0 ∈ (0...0)) | |
14 | 12, 13 | ax-mp 5 | . . . . 5 ⊢ 0 ∈ (0...0) |
15 | eleq1 2200 | . . . . 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 8827 | . . . . . 6 ⊢ (0 + 1) = 1 | |
19 | 18 | oveq1i 5777 | . . . . 5 ⊢ ((0 + 1)...1) = (1...1) |
20 | 19 | eleq2i 2204 | . . . 4 ⊢ (𝐴 ∈ ((0 + 1)...1) ↔ 𝐴 ∈ (1...1)) |
21 | elfz1eq 9808 | . . . . 5 ⊢ (𝐴 ∈ (1...1) → 𝐴 = 1) | |
22 | 1nn 8724 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
23 | nnuz 9354 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
24 | 22, 23 | eleqtri 2212 | . . . . . . 7 ⊢ 1 ∈ (ℤ≥‘1) |
25 | eluzfz1 9804 | . . . . . . 7 ⊢ (1 ∈ (ℤ≥‘1) → 1 ∈ (1...1)) | |
26 | 24, 25 | ax-mp 5 | . . . . . 6 ⊢ 1 ∈ (1...1) |
27 | eleq1 2200 | . . . . . 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 753 | . 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 697 = wceq 1331 ∈ wcel 1480 ∪ cun 3064 ‘cfv 5118 (class class class)co 5767 0cc0 7613 1c1 7614 + caddc 7616 ℕcn 8713 ℕ0cn0 8970 ℤ≥cuz 9319 ...cfz 9783 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 |
This theorem is referenced by: hashfiv01gt1 10521 mod2eq1n2dvds 11565 |
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