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Mirrors > Home > ILE Home > Th. List > fzp1disj | 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 10064 | . . 3 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → (𝑁 + 1) ≤ 𝑁) | |
2 | elfzel2 10059 | . . . 4 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) | |
3 | zre 9292 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
4 | 3 | ltp1d 8922 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 < (𝑁 + 1)) |
5 | peano2z 9324 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
6 | zltnle 9334 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁)) | |
7 | 5, 6 | mpdan 421 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁)) |
8 | 4, 7 | mpbid 147 | . . . 4 ⊢ (𝑁 ∈ ℤ → ¬ (𝑁 + 1) ≤ 𝑁) |
9 | 2, 8 | syl 14 | . . 3 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → ¬ (𝑁 + 1) ≤ 𝑁) |
10 | 1, 9 | pm2.65i 640 | . 2 ⊢ ¬ (𝑁 + 1) ∈ (𝑀...𝑁) |
11 | disjsn 3672 | . 2 ⊢ (((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ ¬ (𝑁 + 1) ∈ (𝑀...𝑁)) | |
12 | 10, 11 | mpbir 146 | 1 ⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ∩ cin 3143 ∅c0 3437 {csn 3610 class class class wbr 4021 (class class class)co 5900 1c1 7847 + caddc 7849 < clt 8027 ≤ cle 8028 ℤcz 9288 ...cfz 10044 |
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 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-addcom 7946 ax-addass 7948 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-0id 7954 ax-rnegex 7955 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-ltadd 7962 |
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-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-br 4022 df-opab 4083 df-mpt 4084 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 df-sub 8165 df-neg 8166 df-inn 8955 df-n0 9212 df-z 9289 df-uz 9564 df-fz 10045 |
This theorem is referenced by: fseq1p1m1 10130 frecfzennn 10463 zfz1isolem1 10861 |
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