Theorem nnc0suc 4413

Description: All naturals are either zero or a successor. Theorem X.1.7 of [Rosser] p. 276. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
nnc0suc	⊢ (A ∈ Nn ↔ (A = 0c ∨ ∃x ∈ Nn A = (x +c 1c)))

Distinct variable group:   x,A

Proof of Theorem nnc0suc

Dummy variables n m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sn 3742	. . . . . 6 ⊢ {0c} = {n ∣ n = 0c}
2		vex 2863	. . . . . . . . 9 ⊢ n ∈ V
3	2	elimak 4260	. . . . . . . 8 ⊢ (n ∈ (Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) “k Nn ) ↔ ∃x ∈ Nn ⟪x, n⟫ ∈ Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c))
4		vex 2863	. . . . . . . . . . 11 ⊢ x ∈ V
5		opkelimagekg 4272	. . . . . . . . . . 11 ⊢ ((x ∈ V ∧ n ∈ V) → (⟪x, n⟫ ∈ Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ↔ n = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) “k x)))
6	4, 2, 5	mp2an 653	. . . . . . . . . 10 ⊢ (⟪x, n⟫ ∈ Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ↔ n = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) “k x))
7		dfaddc2 4382	. . . . . . . . . . 11 ⊢ (x +c 1c) = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) “k x)
8	7	eqeq2i 2363	. . . . . . . . . 10 ⊢ (n = (x +c 1c) ↔ n = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) “k x))
9	6, 8	bitr4i 243	. . . . . . . . 9 ⊢ (⟪x, n⟫ ∈ Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ↔ n = (x +c 1c))
10	9	rexbii 2640	. . . . . . . 8 ⊢ (∃x ∈ Nn ⟪x, n⟫ ∈ Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ↔ ∃x ∈ Nn n = (x +c 1c))
11	3, 10	bitri 240	. . . . . . 7 ⊢ (n ∈ (Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) “k Nn ) ↔ ∃x ∈ Nn n = (x +c 1c))
12	11	eqabi 2465	. . . . . 6 ⊢ (Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) “k Nn ) = {n ∣ ∃x ∈ Nn n = (x +c 1c)}
13	1, 12	uneq12i 3417	. . . . 5 ⊢ ({0c} ∪ (Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) “k Nn )) = ({n ∣ n = 0c} ∪ {n ∣ ∃x ∈ Nn n = (x +c 1c)})
14		unab 3522	. . . . 5 ⊢ ({n ∣ n = 0c} ∪ {n ∣ ∃x ∈ Nn n = (x +c 1c)}) = {n ∣ (n = 0c ∨ ∃x ∈ Nn n = (x +c 1c))}
15	13, 14	eqtri 2373	. . . 4 ⊢ ({0c} ∪ (Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) “k Nn )) = {n ∣ (n = 0c ∨ ∃x ∈ Nn n = (x +c 1c))}
16		snex 4112	. . . . 5 ⊢ {0c} ∈ V
17		addcexlem 4383	. . . . . . . 8 ⊢ ( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) ∈ V
18		1cex 4143	. . . . . . . . . 10 ⊢ 1c ∈ V
19	18	pw1ex 4304	. . . . . . . . 9 ⊢ ℘11c ∈ V
20	19	pw1ex 4304	. . . . . . . 8 ⊢ ℘1℘11c ∈ V
21	17, 20	imakex 4301	. . . . . . 7 ⊢ (( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∈ V
22	21	imagekex 4313	. . . . . 6 ⊢ Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∈ V
23		nncex 4397	. . . . . 6 ⊢ Nn ∈ V
24	22, 23	imakex 4301	. . . . 5 ⊢ (Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) “k Nn ) ∈ V
25	16, 24	unex 4107	. . . 4 ⊢ ({0c} ∪ (Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) “k Nn )) ∈ V
26	15, 25	eqeltrri 2424	. . 3 ⊢ {n ∣ (n = 0c ∨ ∃x ∈ Nn n = (x +c 1c))} ∈ V
27		eqeq1 2359	. . . 4 ⊢ (n = 0c → (n = 0c ↔ 0c = 0c))
28		eqeq1 2359	. . . . 5 ⊢ (n = 0c → (n = (x +c 1c) ↔ 0c = (x +c 1c)))
29	28	rexbidv 2636	. . . 4 ⊢ (n = 0c → (∃x ∈ Nn n = (x +c 1c) ↔ ∃x ∈ Nn 0c = (x +c 1c)))
30	27, 29	orbi12d 690	. . 3 ⊢ (n = 0c → ((n = 0c ∨ ∃x ∈ Nn n = (x +c 1c)) ↔ (0c = 0c ∨ ∃x ∈ Nn 0c = (x +c 1c))))
31		eqeq1 2359	. . . 4 ⊢ (n = m → (n = 0c ↔ m = 0c))
32		eqeq1 2359	. . . . 5 ⊢ (n = m → (n = (x +c 1c) ↔ m = (x +c 1c)))
33	32	rexbidv 2636	. . . 4 ⊢ (n = m → (∃x ∈ Nn n = (x +c 1c) ↔ ∃x ∈ Nn m = (x +c 1c)))
34	31, 33	orbi12d 690	. . 3 ⊢ (n = m → ((n = 0c ∨ ∃x ∈ Nn n = (x +c 1c)) ↔ (m = 0c ∨ ∃x ∈ Nn m = (x +c 1c))))
35		eqeq1 2359	. . . 4 ⊢ (n = (m +c 1c) → (n = 0c ↔ (m +c 1c) = 0c))
36		eqeq1 2359	. . . . 5 ⊢ (n = (m +c 1c) → (n = (x +c 1c) ↔ (m +c 1c) = (x +c 1c)))
37	36	rexbidv 2636	. . . 4 ⊢ (n = (m +c 1c) → (∃x ∈ Nn n = (x +c 1c) ↔ ∃x ∈ Nn (m +c 1c) = (x +c 1c)))
38	35, 37	orbi12d 690	. . 3 ⊢ (n = (m +c 1c) → ((n = 0c ∨ ∃x ∈ Nn n = (x +c 1c)) ↔ ((m +c 1c) = 0c ∨ ∃x ∈ Nn (m +c 1c) = (x +c 1c))))
39		eqeq1 2359	. . . 4 ⊢ (n = A → (n = 0c ↔ A = 0c))
40		eqeq1 2359	. . . . 5 ⊢ (n = A → (n = (x +c 1c) ↔ A = (x +c 1c)))
41	40	rexbidv 2636	. . . 4 ⊢ (n = A → (∃x ∈ Nn n = (x +c 1c) ↔ ∃x ∈ Nn A = (x +c 1c)))
42	39, 41	orbi12d 690	. . 3 ⊢ (n = A → ((n = 0c ∨ ∃x ∈ Nn n = (x +c 1c)) ↔ (A = 0c ∨ ∃x ∈ Nn A = (x +c 1c))))
43		eqid 2353	. . . 4 ⊢ 0c = 0c
44	43	orci 379	. . 3 ⊢ (0c = 0c ∨ ∃x ∈ Nn 0c = (x +c 1c))
45		eqid 2353	. . . . . 6 ⊢ (m +c 1c) = (m +c 1c)
46		addceq1 4384	. . . . . . . 8 ⊢ (x = m → (x +c 1c) = (m +c 1c))
47	46	eqeq2d 2364	. . . . . . 7 ⊢ (x = m → ((m +c 1c) = (x +c 1c) ↔ (m +c 1c) = (m +c 1c)))
48	47	rspcev 2956	. . . . . 6 ⊢ ((m ∈ Nn ∧ (m +c 1c) = (m +c 1c)) → ∃x ∈ Nn (m +c 1c) = (x +c 1c))
49	45, 48	mpan2 652	. . . . 5 ⊢ (m ∈ Nn → ∃x ∈ Nn (m +c 1c) = (x +c 1c))
50	49	olcd 382	. . . 4 ⊢ (m ∈ Nn → ((m +c 1c) = 0c ∨ ∃x ∈ Nn (m +c 1c) = (x +c 1c)))
51	50	a1d 22	. . 3 ⊢ (m ∈ Nn → ((m = 0c ∨ ∃x ∈ Nn m = (x +c 1c)) → ((m +c 1c) = 0c ∨ ∃x ∈ Nn (m +c 1c) = (x +c 1c))))
52	26, 30, 34, 38, 42, 44, 51	finds 4412	. 2 ⊢ (A ∈ Nn → (A = 0c ∨ ∃x ∈ Nn A = (x +c 1c)))
53		peano1 4403	. . . 4 ⊢ 0c ∈ Nn
54		eleq1 2413	. . . 4 ⊢ (A = 0c → (A ∈ Nn ↔ 0c ∈ Nn ))
55	53, 54	mpbiri 224	. . 3 ⊢ (A = 0c → A ∈ Nn )
56		peano2 4404	. . . . 5 ⊢ (x ∈ Nn → (x +c 1c) ∈ Nn )
57		eleq1 2413	. . . . 5 ⊢ (A = (x +c 1c) → (A ∈ Nn ↔ (x +c 1c) ∈ Nn ))
58	56, 57	syl5ibrcom 213	. . . 4 ⊢ (x ∈ Nn → (A = (x +c 1c) → A ∈ Nn ))
59	58	rexlimiv 2733	. . 3 ⊢ (∃x ∈ Nn A = (x +c 1c) → A ∈ Nn )
60	55, 59	jaoi 368	. 2 ⊢ ((A = 0c ∨ ∃x ∈ Nn A = (x +c 1c)) → A ∈ Nn )
61	52, 60	impbii 180	1 ⊢ (A ∈ Nn ↔ (A = 0c ∨ ∃x ∈ Nn A = (x +c 1c)))

Syntax hints:   ↔ wb 176   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209   ⊕ csymdif 3210  {csn 3738  ⟪copk 4058  1_cc1c 4135  ℘₁cpw1 4136   Ins2_k cins2k 4177   Ins3_k cins3k 4178   “_k cimak 4180   SI_k csik 4182  Image_kcimagek 4183   S_k cssetk 4184   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-0c 4378  df-addc 4379  df-nnc 4380

This theorem is referenced by:  nndisjeq  4430  lefinlteq  4464  evenodddisj  4517  sfinltfin  4536  phialllem1  4617  nnc3n3p1  6279