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Mirrors > Home > MPE Home > Th. List > fzp1disj | Structured version Visualization version GIF version |
Description: (𝑀...(𝑁 + 1)) is the disjoint union of (𝑀...𝑁) with {(𝑁 + 1)}. (Contributed by Mario Carneiro, 7-Mar-2014.) |
Ref | Expression |
---|---|
fzp1disj | ⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzle2 13511 | . . 3 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → (𝑁 + 1) ≤ 𝑁) | |
2 | elfzel2 13505 | . . . . 5 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) | |
3 | 2 | zred 12670 | . . . 4 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ ℝ) |
4 | ltp1 12058 | . . . . 5 ⊢ (𝑁 ∈ ℝ → 𝑁 < (𝑁 + 1)) | |
5 | peano2re 11391 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ) | |
6 | ltnle 11297 | . . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ) → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁)) | |
7 | 5, 6 | mpdan 684 | . . . . 5 ⊢ (𝑁 ∈ ℝ → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁)) |
8 | 4, 7 | mpbid 231 | . . . 4 ⊢ (𝑁 ∈ ℝ → ¬ (𝑁 + 1) ≤ 𝑁) |
9 | 3, 8 | syl 17 | . . 3 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → ¬ (𝑁 + 1) ≤ 𝑁) |
10 | 1, 9 | pm2.65i 193 | . 2 ⊢ ¬ (𝑁 + 1) ∈ (𝑀...𝑁) |
11 | disjsn 4710 | . 2 ⊢ (((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ ¬ (𝑁 + 1) ∈ (𝑀...𝑁)) | |
12 | 10, 11 | mpbir 230 | 1 ⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∩ cin 3942 ∅c0 4317 {csn 4623 class class class wbr 5141 (class class class)co 7405 ℝcr 11111 1c1 11113 + caddc 11115 < clt 11252 ≤ cle 11253 ...cfz 13490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-z 12563 df-uz 12827 df-fz 13491 |
This theorem is referenced by: fzdifsuc 13567 fseq1p1m1 13581 fzennn 13939 gsummptfzsplit 19852 telgsumfzslem 19908 imasdsf1olem 24234 wlkp1 29447 esumfzf 33597 subfacp1lem6 34704 mapfzcons 42032 mapfzcons1 42033 mapfzcons2 42035 sge0p1 45702 |
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