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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 3972 | . 2 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ (ℤ ∖ ℕ))) | |
2 | eldif 3972 | . . . . 5 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) ↔ (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ)) | |
3 | elznn 12626 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ ↔ (𝐴 ∈ ℝ ∧ (𝐴 ∈ ℕ ∨ -𝐴 ∈ ℕ0))) | |
4 | 3 | simprbi 496 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ ℕ ∨ -𝐴 ∈ ℕ0)) |
5 | 4 | orcanai 1004 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ) → -𝐴 ∈ ℕ0) |
6 | negneg 11556 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) | |
7 | 6 | adantr 480 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℕ0) → --𝐴 = 𝐴) |
8 | nn0negz 12652 | . . . . . . . . . 10 ⊢ (-𝐴 ∈ ℕ0 → --𝐴 ∈ ℤ) | |
9 | 8 | adantl 481 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℕ0) → --𝐴 ∈ ℤ) |
10 | 7, 9 | eqeltrrd 2839 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℕ0) → 𝐴 ∈ ℤ) |
11 | 10 | ex 412 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)) |
12 | nngt0 12294 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | |
13 | nnre 12270 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
14 | 13 | lt0neg2d 11830 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (0 < 𝐴 ↔ -𝐴 < 0)) |
15 | 12, 14 | mpbid 232 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → -𝐴 < 0) |
16 | 13 | renegcld 11687 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → -𝐴 ∈ ℝ) |
17 | 0re 11260 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
18 | ltnle 11337 | . . . . . . . . . 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 232 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 0 ≤ -𝐴) |
21 | nn0ge0 12548 | . . . . . . . 8 ⊢ (-𝐴 ∈ ℕ0 → 0 ≤ -𝐴) | |
22 | 20, 21 | nsyl3 138 | . . . . . . 7 ⊢ (-𝐴 ∈ ℕ0 → ¬ 𝐴 ∈ ℕ) |
23 | 11, 22 | jca2 513 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℕ0 → (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ))) |
24 | 5, 23 | impbid2 226 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ) ↔ -𝐴 ∈ ℕ0)) |
25 | 2, 24 | bitrid 283 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ (ℤ ∖ ℕ) ↔ -𝐴 ∈ ℕ0)) |
26 | 25 | notbid 318 | . . 3 ⊢ (𝐴 ∈ ℂ → (¬ 𝐴 ∈ (ℤ ∖ ℕ) ↔ ¬ -𝐴 ∈ ℕ0)) |
27 | 26 | pm5.32i 574 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) |
28 | 1, 27 | bitri 275 | 1 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ∖ cdif 3959 class class class wbr 5147 ℂcc 11150 ℝcr 11151 0cc0 11152 < clt 11292 ≤ cle 11293 -cneg 11490 ℕcn 12263 ℕ0cn0 12523 ℤcz 12610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 |
This theorem is referenced by: dmgmaddn0 27080 dmlogdmgm 27081 dmgmaddnn0 27084 lgamgulmlem1 27086 lgamucov 27095 |
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