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Mirrors > Home > MPE Home > Th. List > elfzlmr | Structured version Visualization version GIF version |
Description: A member of a finite interval of integers is either its lower bound or its upper bound or an element of its interior. (Contributed by AV, 5-Feb-2021.) |
Ref | Expression |
---|---|
elfzlmr | ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)..^𝑁) ∨ 𝐾 = 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz2 13546 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | fzpred 13589 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) | |
3 | 2 | eleq2d 2815 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾 ∈ ({𝑀} ∪ ((𝑀 + 1)...𝑁)))) |
4 | elsni 4649 | . . . . 5 ⊢ (𝐾 ∈ {𝑀} → 𝐾 = 𝑀) | |
5 | elfzr 13785 | . . . . 5 ⊢ (𝐾 ∈ ((𝑀 + 1)...𝑁) → (𝐾 ∈ ((𝑀 + 1)..^𝑁) ∨ 𝐾 = 𝑁)) | |
6 | 4, 5 | orim12i 906 | . . . 4 ⊢ ((𝐾 ∈ {𝑀} ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)) → (𝐾 = 𝑀 ∨ (𝐾 ∈ ((𝑀 + 1)..^𝑁) ∨ 𝐾 = 𝑁))) |
7 | elun 4149 | . . . 4 ⊢ (𝐾 ∈ ({𝑀} ∪ ((𝑀 + 1)...𝑁)) ↔ (𝐾 ∈ {𝑀} ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))) | |
8 | 3orass 1087 | . . . 4 ⊢ ((𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)..^𝑁) ∨ 𝐾 = 𝑁) ↔ (𝐾 = 𝑀 ∨ (𝐾 ∈ ((𝑀 + 1)..^𝑁) ∨ 𝐾 = 𝑁))) | |
9 | 6, 7, 8 | 3imtr4i 291 | . . 3 ⊢ (𝐾 ∈ ({𝑀} ∪ ((𝑀 + 1)...𝑁)) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)..^𝑁) ∨ 𝐾 = 𝑁)) |
10 | 3, 9 | biimtrdi 252 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾 ∈ (𝑀...𝑁) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)..^𝑁) ∨ 𝐾 = 𝑁))) |
11 | 1, 10 | mpcom 38 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)..^𝑁) ∨ 𝐾 = 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 845 ∨ w3o 1083 = wceq 1533 ∈ wcel 2098 ∪ cun 3947 {csn 4632 ‘cfv 6553 (class class class)co 7426 1c1 11147 + caddc 11149 ℤ≥cuz 12860 ...cfz 13524 ..^cfzo 13667 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 |
This theorem is referenced by: elfz0lmr 13787 fzone1 32589 |
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