Theorem nn1m1nn 9257

Description: Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)

Assertion

Ref	Expression
nn1m1nn	|- ( A e. NN -> ( A = 1 \/ ( A - 1 ) e. NN ) )

Proof of Theorem nn1m1nn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 720	. . 3 |- ( x = 1 -> ( x = 1 \/ ( x - 1 ) e. NN ) )
2		1cnd 8292	. . 3 |- ( x = 1 -> 1 e. CC )
3	1, 2	2thd 175	. 2 |- ( x = 1 -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> 1 e. CC ) )
4		eqeq1 2241	. . 3 |- ( x = y -> ( x = 1 <-> y = 1 ) )
5		oveq1 6059	. . . 4 |- ( x = y -> ( x - 1 ) = ( y - 1 ) )
6	5	eleq1d 2303	. . 3 |- ( x = y -> ( ( x - 1 ) e. NN <-> ( y - 1 ) e. NN ) )
7	4, 6	orbi12d 801	. 2 |- ( x = y -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( y = 1 \/ ( y - 1 ) e. NN ) ) )
8		eqeq1 2241	. . 3 |- ( x = ( y + 1 ) -> ( x = 1 <-> ( y + 1 ) = 1 ) )
9		oveq1 6059	. . . 4 |- ( x = ( y + 1 ) -> ( x - 1 ) = ( ( y + 1 ) - 1 ) )
10	9	eleq1d 2303	. . 3 |- ( x = ( y + 1 ) -> ( ( x - 1 ) e. NN <-> ( ( y + 1 ) - 1 ) e. NN ) )
11	8, 10	orbi12d 801	. 2 |- ( x = ( y + 1 ) -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( ( y + 1 ) = 1 \/ ( ( y + 1 ) - 1 ) e. NN ) ) )
12		eqeq1 2241	. . 3 |- ( x = A -> ( x = 1 <-> A = 1 ) )
13		oveq1 6059	. . . 4 |- ( x = A -> ( x - 1 ) = ( A - 1 ) )
14	13	eleq1d 2303	. . 3 |- ( x = A -> ( ( x - 1 ) e. NN <-> ( A - 1 ) e. NN ) )
15	12, 14	orbi12d 801	. 2 |- ( x = A -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( A = 1 \/ ( A - 1 ) e. NN ) ) )
16		ax-1cn 8222	. 2 |- 1 e. CC
17		nncn 9247	. . . . . 6 |- ( y e. NN -> y e. CC )
18		pncan 8481	. . . . . 6 |- ( ( y e. CC /\ 1 e. CC ) -> ( ( y + 1 ) - 1 ) = y )
19	17, 16, 18	sylancl 413	. . . . 5 |- ( y e. NN -> ( ( y + 1 ) - 1 ) = y )
20		id 19	. . . . 5 |- ( y e. NN -> y e. NN )
21	19, 20	eqeltrd 2311	. . . 4 |- ( y e. NN -> ( ( y + 1 ) - 1 ) e. NN )
22	21	olcd 742	. . 3 |- ( y e. NN -> ( ( y + 1 ) = 1 \/ ( ( y + 1 ) - 1 ) e. NN ) )
23	22	a1d 22	. 2 |- ( y e. NN -> ( ( y = 1 \/ ( y - 1 ) e. NN ) -> ( ( y + 1 ) = 1 \/ ( ( y + 1 ) - 1 ) e. NN ) ) )
24	3, 7, 11, 15, 16, 23	nnind 9255	1 |- ( A e. NN -> ( A = 1 \/ ( A - 1 ) e. NN ) )

Syntax hints:   -> wi 4   \/ wo 716   = wceq 1398   e. wcel 2205  (class class class)co 6052   CCcc 8127   1c1 8130   + caddc 8132   - cmin 8446   NNcn 9239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-sub 8448  df-inn 9240

This theorem is referenced by:  nn1suc  9258  nnsub  9278  nnm1nn0  9539  nn0ge2m1nn  9562  ballotfilemfc0  13153  ballotfilemfcc  13154