![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > fzouzdisj | GIF version |
Description: A half-open integer range does not overlap the upper integer range starting at the endpoint of the first range. (Contributed by Mario Carneiro, 21-Sep-2016.) |
Ref | Expression |
---|---|
fzouzdisj | ⊢ ((𝐴..^𝐵) ∩ (ℤ≥‘𝐵)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0 3456 | . 2 ⊢ (((𝐴..^𝐵) ∩ (ℤ≥‘𝐵)) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ ((𝐴..^𝐵) ∩ (ℤ≥‘𝐵))) | |
2 | elfzolt2 10188 | . . . . 5 ⊢ (𝑥 ∈ (𝐴..^𝐵) → 𝑥 < 𝐵) | |
3 | 2 | adantr 276 | . . . 4 ⊢ ((𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) → 𝑥 < 𝐵) |
4 | eluzle 9571 | . . . . . 6 ⊢ (𝑥 ∈ (ℤ≥‘𝐵) → 𝐵 ≤ 𝑥) | |
5 | 4 | adantl 277 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) → 𝐵 ≤ 𝑥) |
6 | eluzel2 9564 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘𝐵) → 𝐵 ∈ ℤ) | |
7 | 6 | adantl 277 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) → 𝐵 ∈ ℤ) |
8 | 7 | zred 9406 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) → 𝐵 ∈ ℝ) |
9 | eluzelre 9569 | . . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘𝐵) → 𝑥 ∈ ℝ) | |
10 | 9 | adantl 277 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) → 𝑥 ∈ ℝ) |
11 | 8, 10 | lenltd 8106 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) → (𝐵 ≤ 𝑥 ↔ ¬ 𝑥 < 𝐵)) |
12 | 5, 11 | mpbid 147 | . . . 4 ⊢ ((𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) → ¬ 𝑥 < 𝐵) |
13 | 3, 12 | pm2.65i 640 | . . 3 ⊢ ¬ (𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) |
14 | elin 3333 | . . 3 ⊢ (𝑥 ∈ ((𝐴..^𝐵) ∩ (ℤ≥‘𝐵)) ↔ (𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵))) | |
15 | 13, 14 | mtbir 672 | . 2 ⊢ ¬ 𝑥 ∈ ((𝐴..^𝐵) ∩ (ℤ≥‘𝐵)) |
16 | 1, 15 | mpgbir 1464 | 1 ⊢ ((𝐴..^𝐵) ∩ (ℤ≥‘𝐵)) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ∩ cin 3143 ∅c0 3437 class class class wbr 4018 ‘cfv 5235 (class class class)co 5897 ℝcr 7841 < clt 8023 ≤ cle 8024 ℤcz 9284 ℤ≥cuz 9559 ..^cfzo 10174 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-ltadd 7958 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-inn 8951 df-n0 9208 df-z 9285 df-uz 9560 df-fz 10041 df-fzo 10175 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |