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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 3550 | . 2 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ (ℤ ∖ ℕ))) | |
2 | eldif 3550 | . . . . 5 ⊢ (𝐴 ∈ (ℤ ∖ ℕ) ↔ (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ)) | |
3 | elznn 11270 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ ↔ (𝐴 ∈ ℝ ∧ (𝐴 ∈ ℕ ∨ -𝐴 ∈ ℕ0))) | |
4 | 3 | simprbi 479 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ ℕ ∨ -𝐴 ∈ ℕ0)) |
5 | 4 | orcanai 950 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ) → -𝐴 ∈ ℕ0) |
6 | negneg 10210 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) | |
7 | 6 | adantr 480 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℕ0) → --𝐴 = 𝐴) |
8 | nn0negz 11292 | . . . . . . . . . 10 ⊢ (-𝐴 ∈ ℕ0 → --𝐴 ∈ ℤ) | |
9 | 8 | adantl 481 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℕ0) → --𝐴 ∈ ℤ) |
10 | 7, 9 | eqeltrrd 2689 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℕ0) → 𝐴 ∈ ℤ) |
11 | 10 | ex 449 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)) |
12 | nngt0 10926 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴) | |
13 | nnre 10904 | . . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
14 | 13 | lt0neg2d 10477 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (0 < 𝐴 ↔ -𝐴 < 0)) |
15 | 12, 14 | mpbid 221 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → -𝐴 < 0) |
16 | 13 | renegcld 10336 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → -𝐴 ∈ ℝ) |
17 | 0re 9919 | . . . . . . . . . . 11 ⊢ 0 ∈ ℝ | |
18 | ltnle 9996 | . . . . . . . . . . 11 ⊢ ((-𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝐴 < 0 ↔ ¬ 0 ≤ -𝐴)) | |
19 | 16, 17, 18 | sylancl 693 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (-𝐴 < 0 ↔ ¬ 0 ≤ -𝐴)) |
20 | 15, 19 | mpbid 221 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ¬ 0 ≤ -𝐴) |
21 | nn0ge0 11195 | . . . . . . . . 9 ⊢ (-𝐴 ∈ ℕ0 → 0 ≤ -𝐴) | |
22 | 20, 21 | nsyl3 132 | . . . . . . . 8 ⊢ (-𝐴 ∈ ℕ0 → ¬ 𝐴 ∈ ℕ) |
23 | 22 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℕ0 → ¬ 𝐴 ∈ ℕ)) |
24 | 11, 23 | jcad 554 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℕ0 → (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ))) |
25 | 5, 24 | impbid2 215 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ ℕ) ↔ -𝐴 ∈ ℕ0)) |
26 | 2, 25 | syl5bb 271 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ (ℤ ∖ ℕ) ↔ -𝐴 ∈ ℕ0)) |
27 | 26 | notbid 307 | . . 3 ⊢ (𝐴 ∈ ℂ → (¬ 𝐴 ∈ (ℤ ∖ ℕ) ↔ ¬ -𝐴 ∈ ℕ0)) |
28 | 27 | pm5.32i 667 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) |
29 | 1, 28 | bitri 263 | 1 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 class class class wbr 4583 ℂcc 9813 ℝcr 9814 0cc0 9815 < clt 9953 ≤ cle 9954 -cneg 10146 ℕcn 10897 ℕ0cn0 11169 ℤcz 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 |
This theorem is referenced by: dmgmaddn0 24549 dmlogdmgm 24550 dmgmaddnn0 24553 lgamgulmlem1 24555 lgamucov 24564 |
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