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Mirrors > Home > MPE Home > Th. List > fzouzdisj | Structured version Visualization version 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 | elfzolt2 13042 | . . . . 5 ⊢ (𝑥 ∈ (𝐴..^𝐵) → 𝑥 < 𝐵) | |
2 | 1 | adantr 484 | . . . 4 ⊢ ((𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) → 𝑥 < 𝐵) |
3 | eluzel2 12236 | . . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘𝐵) → 𝐵 ∈ ℤ) | |
4 | 3 | adantl 485 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) → 𝐵 ∈ ℤ) |
5 | 4 | zred 12075 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) → 𝐵 ∈ ℝ) |
6 | eluzelre 12242 | . . . . . 6 ⊢ (𝑥 ∈ (ℤ≥‘𝐵) → 𝑥 ∈ ℝ) | |
7 | 6 | adantl 485 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) → 𝑥 ∈ ℝ) |
8 | eluzle 12244 | . . . . . 6 ⊢ (𝑥 ∈ (ℤ≥‘𝐵) → 𝐵 ≤ 𝑥) | |
9 | 8 | adantl 485 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) → 𝐵 ≤ 𝑥) |
10 | 5, 7, 9 | lensymd 10780 | . . . 4 ⊢ ((𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) → ¬ 𝑥 < 𝐵) |
11 | 2, 10 | pm2.65i 197 | . . 3 ⊢ ¬ (𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵)) |
12 | elin 3897 | . . 3 ⊢ (𝑥 ∈ ((𝐴..^𝐵) ∩ (ℤ≥‘𝐵)) ↔ (𝑥 ∈ (𝐴..^𝐵) ∧ 𝑥 ∈ (ℤ≥‘𝐵))) | |
13 | 11, 12 | mtbir 326 | . 2 ⊢ ¬ 𝑥 ∈ ((𝐴..^𝐵) ∩ (ℤ≥‘𝐵)) |
14 | 13 | nel0 4264 | 1 ⊢ ((𝐴..^𝐵) ∩ (ℤ≥‘𝐵)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ∅c0 4243 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 < clt 10664 ≤ cle 10665 ℤcz 11969 ℤ≥cuz 12231 ..^cfzo 13028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 |
This theorem is referenced by: bitsres 15812 sseqfv1 31757 sseqfn 31758 sseqf 31760 sseqfv2 31762 |
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