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Mirrors > Home > MPE Home > Th. List > eldmgm | Structured version Visualization version GIF version |
Description: Elementhood in the set of non-nonpositive integers. (Contributed by Mario Carneiro, 12-Jul-2014.) |
Ref | Expression |
---|---|
eldmgm | ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3943 | . 2 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ (ℤ ∖ ℕ))) | |
2 | eldif 3943 | . . . . 5 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) ↔ (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ)) | |
3 | elznn 11985 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ ↔ (𝐴 ∈ ℝ ∧ (𝐴 ∈ ℕ ∨ -𝐴 ∈ ℕ0))) | |
4 | 3 | simprbi 497 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ ℕ ∨ -𝐴 ∈ ℕ0)) |
5 | 4 | orcanai 996 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ) → -𝐴 ∈ ℕ0) |
6 | negneg 10924 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) | |
7 | 6 | adantr 481 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℕ0) → --𝐴 = 𝐴) |
8 | nn0negz 12008 | . . . . . . . . . 10 ⊢ (-𝐴 ∈ ℕ0 → --𝐴 ∈ ℤ) | |
9 | 8 | adantl 482 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℕ0) → --𝐴 ∈ ℤ) |
10 | 7, 9 | eqeltrrd 2911 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℕ0) → 𝐴 ∈ ℤ) |
11 | 10 | ex 413 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)) |
12 | nngt0 11656 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | |
13 | nnre 11633 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
14 | 13 | lt0neg2d 11198 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (0 < 𝐴 ↔ -𝐴 < 0)) |
15 | 12, 14 | mpbid 233 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → -𝐴 < 0) |
16 | 13 | renegcld 11055 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → -𝐴 ∈ ℝ) |
17 | 0re 10631 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
18 | ltnle 10708 | . . . . . . . . . 10 ⊢ ((-𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝐴 < 0 ↔ ¬ 0 ≤ -𝐴)) | |
19 | 16, 17, 18 | sylancl 586 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (-𝐴 < 0 ↔ ¬ 0 ≤ -𝐴)) |
20 | 15, 19 | mpbid 233 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 0 ≤ -𝐴) |
21 | nn0ge0 11910 | . . . . . . . 8 ⊢ (-𝐴 ∈ ℕ0 → 0 ≤ -𝐴) | |
22 | 20, 21 | nsyl3 140 | . . . . . . 7 ⊢ (-𝐴 ∈ ℕ0 → ¬ 𝐴 ∈ ℕ) |
23 | 11, 22 | jca2 514 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℕ0 → (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ))) |
24 | 5, 23 | impbid2 227 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ) ↔ -𝐴 ∈ ℕ0)) |
25 | 2, 24 | syl5bb 284 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ (ℤ ∖ ℕ) ↔ -𝐴 ∈ ℕ0)) |
26 | 25 | notbid 319 | . . 3 ⊢ (𝐴 ∈ ℂ → (¬ 𝐴 ∈ (ℤ ∖ ℕ) ↔ ¬ -𝐴 ∈ ℕ0)) |
27 | 26 | pm5.32i 575 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) |
28 | 1, 27 | bitri 276 | 1 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 class class class wbr 5057 ℂcc 10523 ℝcr 10524 0cc0 10525 < clt 10663 ≤ cle 10664 -cneg 10859 ℕcn 11626 ℕ0cn0 11885 ℤcz 11969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 |
This theorem is referenced by: dmgmaddn0 25527 dmlogdmgm 25528 dmgmaddnn0 25531 lgamgulmlem1 25533 lgamucov 25542 |
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