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Mirrors > Home > MPE Home > Th. List > elznn0nn | Structured version Visualization version GIF version |
Description: Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.) |
Ref | Expression |
---|---|
elznn0nn | ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 12183 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
2 | andi 1008 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ) ∨ -𝑁 ∈ ℕ)) ↔ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) | |
3 | df-3or 1090 | . . . 4 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ↔ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ) ∨ -𝑁 ∈ ℕ)) | |
4 | 3 | anbi2i 626 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ (𝑁 ∈ ℝ ∧ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ) ∨ -𝑁 ∈ ℕ))) |
5 | nn0re 12104 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | 5 | pm4.71ri 564 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℕ0)) |
7 | elnn0 12097 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
8 | orcom 870 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) | |
9 | 7, 8 | bitri 278 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) |
10 | 9 | anbi2i 626 | . . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ 𝑁 ∈ ℕ0) ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ))) |
11 | 6, 10 | bitri 278 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ))) |
12 | 11 | orbi1i 914 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) ↔ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
13 | 2, 4, 12 | 3bitr4i 306 | . 2 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
14 | 1, 13 | bitri 278 | 1 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∨ wo 847 ∨ w3o 1088 = wceq 1543 ∈ wcel 2110 ℝcr 10733 0cc0 10734 -cneg 11068 ℕcn 11835 ℕ0cn0 12095 ℤcz 12181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5197 ax-nul 5204 ax-pr 5327 ax-un 7528 ax-1cn 10792 ax-icn 10793 ax-addcl 10794 ax-addrcl 10795 ax-mulcl 10796 ax-mulrcl 10797 ax-i2m1 10802 ax-1ne0 10803 ax-rnegex 10805 ax-rrecex 10806 ax-cnre 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-uni 4825 df-iun 4911 df-br 5059 df-opab 5121 df-mpt 5141 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-pred 6165 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-ov 7221 df-om 7650 df-wrecs 8052 df-recs 8113 df-rdg 8151 df-neg 11070 df-nn 11836 df-n0 12096 df-z 12182 |
This theorem is referenced by: zindd 12283 expcl2lem 13652 mulexpz 13680 expaddz 13684 expmulz 13686 absexpz 14874 bitsfzo 15999 pcid 16431 mulgsubcl 18511 mulgneg 18515 ghmmulg 18639 prmirred 20466 tgpmulg 22995 dvexp3 24880 2sqnn0 26324 ipasslem3 28919 ztprmneprm 45364 |
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