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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 12345 | . . 3 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → (𝑁 + 1) ≤ 𝑁) | |
2 | elfzel2 12340 | . . . . 5 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) | |
3 | 2 | zred 11482 | . . . 4 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ ℝ) |
4 | ltp1 10861 | . . . . 5 ⊢ (𝑁 ∈ ℝ → 𝑁 < (𝑁 + 1)) | |
5 | peano2re 10209 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ) | |
6 | ltnle 10117 | . . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ) → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁)) | |
7 | 5, 6 | mpdan 702 | . . . . 5 ⊢ (𝑁 ∈ ℝ → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁)) |
8 | 4, 7 | mpbid 222 | . . . 4 ⊢ (𝑁 ∈ ℝ → ¬ (𝑁 + 1) ≤ 𝑁) |
9 | 3, 8 | syl 17 | . . 3 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → ¬ (𝑁 + 1) ≤ 𝑁) |
10 | 1, 9 | pm2.65i 185 | . 2 ⊢ ¬ (𝑁 + 1) ∈ (𝑀...𝑁) |
11 | disjsn 4246 | . 2 ⊢ (((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ ¬ (𝑁 + 1) ∈ (𝑀...𝑁)) | |
12 | 10, 11 | mpbir 221 | 1 ⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ∅c0 3915 {csn 4177 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 ...cfz 12326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-z 11378 df-uz 11688 df-fz 12327 |
This theorem is referenced by: fzdifsuc 12400 fseq1p1m1 12414 fzennn 12767 gsummptfzsplit 18332 telgsumfzslem 18385 imasdsf1olem 22178 wlkp1 26578 esumfzf 30131 subfacp1lem6 31167 mapfzcons 37279 mapfzcons1 37280 mapfzcons2 37282 sge0p1 40631 |
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