Theorem nn1m1nn 9322

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 724	. . 3 |- ( x = 1 -> ( x = 1 \/ ( x - 1 ) e. NN ) )
2		1cnd 8342	. . 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 2245	. . 3 |- ( x = y -> ( x = 1 <-> y = 1 ) )
5		oveq1 6092	. . . 4 |- ( x = y -> ( x - 1 ) = ( y - 1 ) )
6	5	eleq1d 2307	. . 3 |- ( x = y -> ( ( x - 1 ) e. NN <-> ( y - 1 ) e. NN ) )
7	4, 6	orbi12d 805	. 2 |- ( x = y -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( y = 1 \/ ( y - 1 ) e. NN ) ) )
8		eqeq1 2245	. . 3 |- ( x = ( y + 1 ) -> ( x = 1 <-> ( y + 1 ) = 1 ) )
9		oveq1 6092	. . . 4 |- ( x = ( y + 1 ) -> ( x - 1 ) = ( ( y + 1 ) - 1 ) )
10	9	eleq1d 2307	. . 3 |- ( x = ( y + 1 ) -> ( ( x - 1 ) e. NN <-> ( ( y + 1 ) - 1 ) e. NN ) )
11	8, 10	orbi12d 805	. 2 |- ( x = ( y + 1 ) -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( ( y + 1 ) = 1 \/ ( ( y + 1 ) - 1 ) e. NN ) ) )
12		eqeq1 2245	. . 3 |- ( x = A -> ( x = 1 <-> A = 1 ) )
13		oveq1 6092	. . . 4 |- ( x = A -> ( x - 1 ) = ( A - 1 ) )
14	13	eleq1d 2307	. . 3 |- ( x = A -> ( ( x - 1 ) e. NN <-> ( A - 1 ) e. NN ) )
15	12, 14	orbi12d 805	. 2 |- ( x = A -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( A = 1 \/ ( A - 1 ) e. NN ) ) )
16		ax-1cn 8272	. 2 |- 1 e. CC
17		nncn 9312	. . . . . 6 |- ( y e. NN -> y e. CC )
18		pncan 8532	. . . . . 6 |- ( ( y e. CC /\ 1 e. CC ) -> ( ( y + 1 ) - 1 ) = y )
19	17, 16, 18	sylancl 417	. . . . 5 |- ( y e. NN -> ( ( y + 1 ) - 1 ) = y )
20		id 19	. . . . 5 |- ( y e. NN -> y e. NN )
21	19, 20	eqeltrd 2315	. . . 4 |- ( y e. NN -> ( ( y + 1 ) - 1 ) e. NN )
22	21	olcd 746	. . 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 9320	1 |- ( A e. NN -> ( A = 1 \/ ( A - 1 ) e. NN ) )

This proof depends on syntax axioms:   -> wi 4   \/ wo 720   = wceq 1402   e. wcel 2209  (class class class)co 6085   CCcc 8177   1c1 8180   + caddc 8182   - cmin 8497   NNcn 9304

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-sub 8499  df-inn 9305

This theorem is used by:  nn1suc  9323  nnsub  9343  nnm1nn0  9604  nn0ge2m1nn  9627  ballotfilemfc0  13232  ballotfilemfcc  13233