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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 9973 | . . 3 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → (𝑁 + 1) ≤ 𝑁) | |
2 | elfzel2 9968 | . . . 4 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) | |
3 | zre 9205 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
4 | 3 | ltp1d 8835 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 < (𝑁 + 1)) |
5 | peano2z 9237 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
6 | zltnle 9247 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁)) | |
7 | 5, 6 | mpdan 419 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁)) |
8 | 4, 7 | mpbid 146 | . . . 4 ⊢ (𝑁 ∈ ℤ → ¬ (𝑁 + 1) ≤ 𝑁) |
9 | 2, 8 | syl 14 | . . 3 ⊢ ((𝑁 + 1) ∈ (𝑀...𝑁) → ¬ (𝑁 + 1) ≤ 𝑁) |
10 | 1, 9 | pm2.65i 634 | . 2 ⊢ ¬ (𝑁 + 1) ∈ (𝑀...𝑁) |
11 | disjsn 3643 | . 2 ⊢ (((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ ¬ (𝑁 + 1) ∈ (𝑀...𝑁)) | |
12 | 10, 11 | mpbir 145 | 1 ⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 = wceq 1348 ∈ wcel 2141 ∩ cin 3120 ∅c0 3414 {csn 3581 class class class wbr 3987 (class class class)co 5851 1c1 7764 + caddc 7766 < clt 7943 ≤ cle 7944 ℤcz 9201 ...cfz 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-addcom 7863 ax-addass 7865 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-0id 7871 ax-rnegex 7872 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-ltadd 7879 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-inn 8868 df-n0 9125 df-z 9202 df-uz 9477 df-fz 9955 |
This theorem is referenced by: fseq1p1m1 10039 frecfzennn 10371 zfz1isolem1 10764 |
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