Theorem elnn0nn 4593

Description: The nonnegative integer property expressed in terms of natural numbers.

Assertion

Ref	Expression
elnn0nn	⊢ (A ∈ ℕ0 ↔ (A ∈ ℂ ∧ (A + 1) ∈ ℕ))

Proof of Theorem elnn0nn

Step	Hyp	Ref	Expression
1		peano2nn 4433	. . . . . . 7 ⊢ (A ∈ ℕ → (A + 1) ∈ ℕ)
2		1nn 4432	. . . . . . . 8 ⊢ 1 ∈ ℕ
3		opreq1 3006	. . . . . . . . . 10 ⊢ (A = 0 → (A + 1) = (0 + 1))
4		1cn 4101	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
5	4	addid2 4113	. . . . . . . . . 10 ⊢ (0 + 1) = 1
6	3, 5	syl6eq 1140	. . . . . . . . 9 ⊢ (A = 0 → (A + 1) = 1)
7	6	eleq1d 1155	. . . . . . . 8 ⊢ (A = 0 → ((A + 1) ∈ ℕ ↔ 1 ∈ ℕ))
8	2, 7	mpbiri 169	. . . . . . 7 ⊢ (A = 0 → (A + 1) ∈ ℕ)
9	1, 8	jaoi 275	. . . . . 6 ⊢ ((A ∈ ℕ ∨ A = 0) → (A + 1) ∈ ℕ)
10	9	a1i 7	. . . . 5 ⊢ (A ∈ ℝ → ((A ∈ ℕ ∨ A = 0) → (A + 1) ∈ ℕ))
11		recnt 4097	. . . . . . . . 9 ⊢ (A ∈ ℝ → A ∈ ℂ)
12		1z 4584	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
13		zrevaddclt 4592	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → ((A ∈ ℂ ∧ (A + 1) ∈ ℤ) ↔ A ∈ ℤ))
14	12, 13	ax-mp 6	. . . . . . . . . . 11 ⊢ ((A ∈ ℂ ∧ (A + 1) ∈ ℤ) ↔ A ∈ ℤ)
15	14	biimp 133	. . . . . . . . . 10 ⊢ ((A ∈ ℂ ∧ (A + 1) ∈ ℤ) → A ∈ ℤ)
16	15	exp 291	. . . . . . . . 9 ⊢ (A ∈ ℂ → ((A + 1) ∈ ℤ → A ∈ ℤ))
17	11, 16	syl 12	. . . . . . . 8 ⊢ (A ∈ ℝ → ((A + 1) ∈ ℤ → A ∈ ℤ))
18		ax1re 4064	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
19		ax0re 4063	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
20		leadd1t 4350	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ A ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ A ↔ (0 + 1) ≤ (A + 1)))
21	19, 20	mp3an1 639	. . . . . . . . . . . 12 ⊢ ((A ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ A ↔ (0 + 1) ≤ (A + 1)))
22	18, 21	mpan2 519	. . . . . . . . . . 11 ⊢ (A ∈ ℝ → (0 ≤ A ↔ (0 + 1) ≤ (A + 1)))
23	5	breq1i 2068	. . . . . . . . . . 11 ⊢ ((0 + 1) ≤ (A + 1) ↔ 1 ≤ (A + 1))
24	22, 23	syl6bb 414	. . . . . . . . . 10 ⊢ (A ∈ ℝ → (0 ≤ A ↔ 1 ≤ (A + 1)))
25		leloet 4284	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ A ∈ ℝ) → (0 ≤ A ↔ (0 < A ∨ 0 = A)))
26	19, 25	mpan 518	. . . . . . . . . 10 ⊢ (A ∈ ℝ → (0 ≤ A ↔ (0 < A ∨ 0 = A)))
27	24, 26	bitr3d 408	. . . . . . . . 9 ⊢ (A ∈ ℝ → (1 ≤ (A + 1) ↔ (0 < A ∨ 0 = A)))
28	27	biimpd 135	. . . . . . . 8 ⊢ (A ∈ ℝ → (1 ≤ (A + 1) → (0 < A ∨ 0 = A)))
29	17, 28	anim12d 431	. . . . . . 7 ⊢ (A ∈ ℝ → (((A + 1) ∈ ℤ ∧ 1 ≤ (A + 1)) → (A ∈ ℤ ∧ (0 < A ∨ 0 = A))))
30		andi 456	. . . . . . . 8 ⊢ ((A ∈ ℤ ∧ (0 < A ∨ 0 = A)) ↔ ((A ∈ ℤ ∧ 0 < A) ∨ (A ∈ ℤ ∧ 0 = A)))
31		elnnz 4572	. . . . . . . . 9 ⊢ (A ∈ ℕ ↔ (A ∈ ℤ ∧ 0 < A))
32		cleqcom 1103	. . . . . . . . . 10 ⊢ (A = 0 ↔ 0 = A)
33		0z 4573	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
34		eleq1 1149	. . . . . . . . . . . 12 ⊢ (0 = A → (0 ∈ ℤ ↔ A ∈ ℤ))
35	33, 34	mpbii 168	. . . . . . . . . . 11 ⊢ (0 = A → A ∈ ℤ)
36	35	pm4.71ri 484	. . . . . . . . . 10 ⊢ (0 = A ↔ (A ∈ ℤ ∧ 0 = A))
37	32, 36	bitr 151	. . . . . . . . 9 ⊢ (A = 0 ↔ (A ∈ ℤ ∧ 0 = A))
38	31, 37	orbi12i 216	. . . . . . . 8 ⊢ ((A ∈ ℕ ∨ A = 0) ↔ ((A ∈ ℤ ∧ 0 < A) ∨ (A ∈ ℤ ∧ 0 = A)))
39	30, 38	bitr4 154	. . . . . . 7 ⊢ ((A ∈ ℤ ∧ (0 < A ∨ 0 = A)) ↔ (A ∈ ℕ ∨ A = 0))
40	29, 39	syl6ib 185	. . . . . 6 ⊢ (A ∈ ℝ → (((A + 1) ∈ ℤ ∧ 1 ≤ (A + 1)) → (A ∈ ℕ ∨ A = 0)))
41		elnnz1 4581	. . . . . 6 ⊢ ((A + 1) ∈ ℕ ↔ ((A + 1) ∈ ℤ ∧ 1 ≤ (A + 1)))
42	40, 41	syl5ib 181	. . . . 5 ⊢ (A ∈ ℝ → ((A + 1) ∈ ℕ → (A ∈ ℕ ∨ A = 0)))
43	10, 42	impbid 397	. . . 4 ⊢ (A ∈ ℝ → ((A ∈ ℕ ∨ A = 0) ↔ (A + 1) ∈ ℕ))
44		elnn0 4536	. . . 4 ⊢ (A ∈ ℕ0 ↔ (A ∈ ℕ ∨ A = 0))
45	43, 44	syl5bb 410	. . 3 ⊢ (A ∈ ℝ → (A ∈ ℕ0 ↔ (A + 1) ∈ ℕ))
46	45	pm5.32i 489	. 2 ⊢ ((A ∈ ℝ ∧ A ∈ ℕ0) ↔ (A ∈ ℝ ∧ (A + 1) ∈ ℕ))
47		nn0ret 4542	. . 3 ⊢ (A ∈ ℕ0 → A ∈ ℝ)
48	47	pm4.71ri 484	. 2 ⊢ (A ∈ ℕ0 ↔ (A ∈ ℝ ∧ A ∈ ℕ0))
49		zret 4567	. . . . . 6 ⊢ (A ∈ ℤ → A ∈ ℝ)
50	14, 49	sylbi 174	. . . . 5 ⊢ ((A ∈ ℂ ∧ (A + 1) ∈ ℤ) → A ∈ ℝ)
51		nnzt 4579	. . . . 5 ⊢ ((A + 1) ∈ ℕ → (A + 1) ∈ ℤ)
52	50, 51	sylan2 346	. . . 4 ⊢ ((A ∈ ℂ ∧ (A + 1) ∈ ℕ) → A ∈ ℝ)
53		pm3.27 260	. . . 4 ⊢ ((A ∈ ℂ ∧ (A + 1) ∈ ℕ) → (A + 1) ∈ ℕ)
54	52, 53	jca 236	. . 3 ⊢ ((A ∈ ℂ ∧ (A + 1) ∈ ℕ) → (A ∈ ℝ ∧ (A + 1) ∈ ℕ))
55	11	anim1i 269	. . 3 ⊢ ((A ∈ ℝ ∧ (A + 1) ∈ ℕ) → (A ∈ ℂ ∧ (A + 1) ∈ ℕ))
56	54, 55	impbi 139	. 2 ⊢ ((A ∈ ℂ ∧ (A + 1) ∈ ℕ) ↔ (A ∈ ℝ ∧ (A + 1) ∈ ℕ))
57	46, 48, 56	3bitr4 158	1 ⊢ (A ∈ ℕ0 ↔ (A ∈ ℂ ∧ (A + 1) ∈ ℕ))

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   class class class wbr 2054  (class class class)co 3001  ℂcc 4026  ℝcr 4027  0cc0 4028  1c1 4029   + caddc 4031   < clt 4033   ≤ cle 4092  ℕcn 4093  ℕ₀cn0 4094  ℤcz 4095

This theorem is referenced by:  elnnnn0 4594  facclt 4874  nn0ennn 4925

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-le 4277  df-n 4423  df-n0 4535  df-z 4564

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