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Mirrors > Home > MPE Home > Th. List > 0nelfz1 | Structured version Visualization version GIF version |
Description: 0 is not an element of a finite interval of integers starting at 1. (Contributed by AV, 27-Aug-2020.) |
Ref | Expression |
---|---|
0nelfz1 | ⊢ 0 ∉ (1...𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lt1 11508 | . . . . 5 ⊢ 0 < 1 | |
2 | 0re 10988 | . . . . . 6 ⊢ 0 ∈ ℝ | |
3 | 1re 10986 | . . . . . 6 ⊢ 1 ∈ ℝ | |
4 | 2, 3 | ltnlei 11107 | . . . . 5 ⊢ (0 < 1 ↔ ¬ 1 ≤ 0) |
5 | 1, 4 | mpbi 229 | . . . 4 ⊢ ¬ 1 ≤ 0 |
6 | 5 | intnanr 488 | . . 3 ⊢ ¬ (1 ≤ 0 ∧ 0 ≤ 𝑁) |
7 | 6 | intnan 487 | . 2 ⊢ ¬ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (1 ≤ 0 ∧ 0 ≤ 𝑁)) |
8 | df-nel 3052 | . . 3 ⊢ (0 ∉ (1...𝑁) ↔ ¬ 0 ∈ (1...𝑁)) | |
9 | elfz2 13257 | . . 3 ⊢ (0 ∈ (1...𝑁) ↔ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (1 ≤ 0 ∧ 0 ≤ 𝑁))) | |
10 | 8, 9 | xchbinx 334 | . 2 ⊢ (0 ∉ (1...𝑁) ↔ ¬ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (1 ≤ 0 ∧ 0 ≤ 𝑁))) |
11 | 7, 10 | mpbir 230 | 1 ⊢ 0 ∉ (1...𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2110 ∉ wnel 3051 class class class wbr 5079 (class class class)co 7272 0cc0 10882 1c1 10883 < clt 11020 ≤ cle 11021 ℤcz 12330 ...cfz 13250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-po 5504 df-so 5505 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-1st 7825 df-2nd 7826 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-z 12331 df-fz 13251 |
This theorem is referenced by: lcmflefac 16364 prmodvdslcmf 16759 prmolelcmf 16760 prmgaplcmlem2 16764 prmgaplcm 16772 f1resfz0f1d 33081 |
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