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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 9369 | . . . . . 6 ⊢ 1 ∈ (ℤ≥‘0) | |
2 | eluzfz1 9811 | . . . . . 6 ⊢ (1 ∈ (ℤ≥‘0) → 0 ∈ (0...1)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 0 ∈ (0...1) |
4 | fzsplit 9831 | . . . . 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 2206 | . . 3 ⊢ (𝐴 ∈ (0...1) ↔ 𝐴 ∈ ((0...0) ∪ ((0 + 1)...1))) |
7 | elun 3217 | . . 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 9815 | . . . 4 ⊢ (𝐴 ∈ (0...0) → 𝐴 = 0) | |
10 | 0nn0 8992 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
11 | nn0uz 9360 | . . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0) | |
12 | 10, 11 | eleqtri 2214 | . . . . . 6 ⊢ 0 ∈ (ℤ≥‘0) |
13 | eluzfz1 9811 | . . . . . 6 ⊢ (0 ∈ (ℤ≥‘0) → 0 ∈ (0...0)) | |
14 | 12, 13 | ax-mp 5 | . . . . 5 ⊢ 0 ∈ (0...0) |
15 | eleq1 2202 | . . . . 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 8834 | . . . . . 6 ⊢ (0 + 1) = 1 | |
19 | 18 | oveq1i 5784 | . . . . 5 ⊢ ((0 + 1)...1) = (1...1) |
20 | 19 | eleq2i 2206 | . . . 4 ⊢ (𝐴 ∈ ((0 + 1)...1) ↔ 𝐴 ∈ (1...1)) |
21 | elfz1eq 9815 | . . . . 5 ⊢ (𝐴 ∈ (1...1) → 𝐴 = 1) | |
22 | 1nn 8731 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
23 | nnuz 9361 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
24 | 22, 23 | eleqtri 2214 | . . . . . . 7 ⊢ 1 ∈ (ℤ≥‘1) |
25 | eluzfz1 9811 | . . . . . . 7 ⊢ (1 ∈ (ℤ≥‘1) → 1 ∈ (1...1)) | |
26 | 24, 25 | ax-mp 5 | . . . . . 6 ⊢ 1 ∈ (1...1) |
27 | eleq1 2202 | . . . . . 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 3069 ‘cfv 5123 (class class class)co 5774 0cc0 7620 1c1 7621 + caddc 7623 ℕcn 8720 ℕ0cn0 8977 ℤ≥cuz 9326 ...cfz 9790 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 |
This theorem is referenced by: hashfiv01gt1 10528 mod2eq1n2dvds 11576 |
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