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 e. Nn <-> (A = 0c \/ E.x e. Nn A = (x +o 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 e. _V
3	2	elimak 4260	. . . . . . . 8 |- (n e. (Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"k Nn ) <-> E.x e. Nn <<x, n>> e. Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c))
4		vex 2863	. . . . . . . . . . 11 |- x e. _V
5		opkelimagekg 4272	. . . . . . . . . . 11 |- ((x e. _V /\ n e. _V) -> (<<x, n>> e. Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) <-> n = ((( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"kx)))
6	4, 2, 5	mp2an 653	. . . . . . . . . 10 |- (<<x, n>> e. Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) <-> n = ((( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"kx))
7		dfaddc2 4382	. . . . . . . . . . 11 |- (x +o 1c) = ((( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"kx)
8	7	eqeq2i 2363	. . . . . . . . . 10 |- (n = (x +o 1c) <-> n = ((( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"kx))
9	6, 8	bitr4i 243	. . . . . . . . 9 |- (<<x, n>> e. Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) <-> n = (x +o 1c))
10	9	rexbii 2640	. . . . . . . 8 |- (E.x e. Nn <<x, n>> e. Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) <-> E.x e. Nn n = (x +o 1c))
11	3, 10	bitri 240	. . . . . . 7 |- (n e. (Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"k Nn ) <-> E.x e. Nn n = (x +o 1c))
12	11	eqabi 2465	. . . . . 6 |- (Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"k Nn ) = {n | E.x e. Nn n = (x +o 1c)}
13	1, 12	uneq12i 3417	. . . . 5 |- ({0c} u. (Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"k Nn )) = ({n | n = 0c} u. {n | E.x e. Nn n = (x +o 1c)})
14		unab 3522	. . . . 5 |- ({n | n = 0c} u. {n | E.x e. Nn n = (x +o 1c)}) = {n | (n = 0c \/ E.x e. Nn n = (x +o 1c))}
15	13, 14	eqtri 2373	. . . 4 |- ({0c} u. (Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"k Nn )) = {n | (n = 0c \/ E.x e. Nn n = (x +o 1c))}
16		snex 4112	. . . . 5 |- {0c} e. _V
17		addcexlem 4383	. . . . . . . 8 |- ( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c)) e. _V
18		1cex 4143	. . . . . . . . . 10 |- 1c e. _V
19	18	pw1ex 4304	. . . . . . . . 9 |- ~P1 1c e. _V
20	19	pw1ex 4304	. . . . . . . 8 |- ~P1 ~P1 1c e. _V
21	17, 20	imakex 4301	. . . . . . 7 |- (( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) e. _V
22	21	imagekex 4313	. . . . . 6 |- Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) e. _V
23		nncex 4397	. . . . . 6 |- Nn e. _V
24	22, 23	imakex 4301	. . . . 5 |- (Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"k Nn ) e. _V
25	16, 24	unex 4107	. . . 4 |- ({0c} u. (Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"k Nn )) e. _V
26	15, 25	eqeltrri 2424	. . 3 |- {n | (n = 0c \/ E.x e. Nn n = (x +o 1c))} e. _V
27		eqeq1 2359	. . . 4 |- (n = 0c -> (n = 0c <-> 0c = 0c))
28		eqeq1 2359	. . . . 5 |- (n = 0c -> (n = (x +o 1c) <-> 0c = (x +o 1c)))
29	28	rexbidv 2636	. . . 4 |- (n = 0c -> (E.x e. Nn n = (x +o 1c) <-> E.x e. Nn 0c = (x +o 1c)))
30	27, 29	orbi12d 690	. . 3 |- (n = 0c -> ((n = 0c \/ E.x e. Nn n = (x +o 1c)) <-> (0c = 0c \/ E.x e. Nn 0c = (x +o 1c))))
31		eqeq1 2359	. . . 4 |- (n = m -> (n = 0c <-> m = 0c))
32		eqeq1 2359	. . . . 5 |- (n = m -> (n = (x +o 1c) <-> m = (x +o 1c)))
33	32	rexbidv 2636	. . . 4 |- (n = m -> (E.x e. Nn n = (x +o 1c) <-> E.x e. Nn m = (x +o 1c)))
34	31, 33	orbi12d 690	. . 3 |- (n = m -> ((n = 0c \/ E.x e. Nn n = (x +o 1c)) <-> (m = 0c \/ E.x e. Nn m = (x +o 1c))))
35		eqeq1 2359	. . . 4 |- (n = (m +o 1c) -> (n = 0c <-> (m +o 1c) = 0c))
36		eqeq1 2359	. . . . 5 |- (n = (m +o 1c) -> (n = (x +o 1c) <-> (m +o 1c) = (x +o 1c)))
37	36	rexbidv 2636	. . . 4 |- (n = (m +o 1c) -> (E.x e. Nn n = (x +o 1c) <-> E.x e. Nn (m +o 1c) = (x +o 1c)))
38	35, 37	orbi12d 690	. . 3 |- (n = (m +o 1c) -> ((n = 0c \/ E.x e. Nn n = (x +o 1c)) <-> ((m +o 1c) = 0c \/ E.x e. Nn (m +o 1c) = (x +o 1c))))
39		eqeq1 2359	. . . 4 |- (n = A -> (n = 0c <-> A = 0c))
40		eqeq1 2359	. . . . 5 |- (n = A -> (n = (x +o 1c) <-> A = (x +o 1c)))
41	40	rexbidv 2636	. . . 4 |- (n = A -> (E.x e. Nn n = (x +o 1c) <-> E.x e. Nn A = (x +o 1c)))
42	39, 41	orbi12d 690	. . 3 |- (n = A -> ((n = 0c \/ E.x e. Nn n = (x +o 1c)) <-> (A = 0c \/ E.x e. Nn A = (x +o 1c))))
43		eqid 2353	. . . 4 |- 0c = 0c
44	43	orci 379	. . 3 |- (0c = 0c \/ E.x e. Nn 0c = (x +o 1c))
45		eqid 2353	. . . . . 6 |- (m +o 1c) = (m +o 1c)
46		addceq1 4384	. . . . . . . 8 |- (x = m -> (x +o 1c) = (m +o 1c))
47	46	eqeq2d 2364	. . . . . . 7 |- (x = m -> ((m +o 1c) = (x +o 1c) <-> (m +o 1c) = (m +o 1c)))
48	47	rspcev 2956	. . . . . 6 |- ((m e. Nn /\ (m +o 1c) = (m +o 1c)) -> E.x e. Nn (m +o 1c) = (x +o 1c))
49	45, 48	mpan2 652	. . . . 5 |- (m e. Nn -> E.x e. Nn (m +o 1c) = (x +o 1c))
50	49	olcd 382	. . . 4 |- (m e. Nn -> ((m +o 1c) = 0c \/ E.x e. Nn (m +o 1c) = (x +o 1c)))
51	50	a1d 22	. . 3 |- (m e. Nn -> ((m = 0c \/ E.x e. Nn m = (x +o 1c)) -> ((m +o 1c) = 0c \/ E.x e. Nn (m +o 1c) = (x +o 1c))))
52	26, 30, 34, 38, 42, 44, 51	finds 4412	. 2 |- (A e. Nn -> (A = 0c \/ E.x e. Nn A = (x +o 1c)))
53		peano1 4403	. . . 4 |- 0c e. Nn
54		eleq1 2413	. . . 4 |- (A = 0c -> (A e. Nn <-> 0c e. Nn ))
55	53, 54	mpbiri 224	. . 3 |- (A = 0c -> A e. Nn )
56		peano2 4404	. . . . 5 |- (x e. Nn -> (x +o 1c) e. Nn )
57		eleq1 2413	. . . . 5 |- (A = (x +o 1c) -> (A e. Nn <-> (x +o 1c) e. Nn ))
58	56, 57	syl5ibrcom 213	. . . 4 |- (x e. Nn -> (A = (x +o 1c) -> A e. Nn ))
59	58	rexlimiv 2733	. . 3 |- (E.x e. Nn A = (x +o 1c) -> A e. Nn )
60	55, 59	jaoi 368	. 2 |- ((A = 0c \/ E.x e. Nn A = (x +o 1c)) -> A e. Nn )
61	52, 60	impbii 180	1 |- (A e. Nn <-> (A = 0c \/ E.x e. Nn A = (x +o 1c)))

This proof depends on syntax axioms:   <-> wb 176   \/ wo 357   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2616  _Vcvv 2860   ∼ ccompl 3206   \ cdif 3207   u. cun 3208   i^i cin 3209   (+) csymdif 3210  {csn 3738  <<copk 4058  1_cc1c 4135  ~P₁ cpw1 4136   Ins2_k cins2k 4177   Ins3_k cins3k 4178  "_kcimak 4180   SI_k csik 4182  Image_kcimagek 4183   S_k cssetk 4184   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376

This proof depends on 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 proof 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 used by:  nndisjeq  4430  lefinlteq  4464  evenodddisj  4517  sfinltfin  4536  phialllem1  4617  nnc3n3p1  6279