Theorem nn1m1nn 9151

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 717	. . 3 |- ( x = 1 -> ( x = 1 \/ ( x - 1 ) e. NN ) )
2		1cnd 8185	. . 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 2236	. . 3 |- ( x = y -> ( x = 1 <-> y = 1 ) )
5		oveq1 6020	. . . 4 |- ( x = y -> ( x - 1 ) = ( y - 1 ) )
6	5	eleq1d 2298	. . 3 |- ( x = y -> ( ( x - 1 ) e. NN <-> ( y - 1 ) e. NN ) )
7	4, 6	orbi12d 798	. 2 |- ( x = y -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( y = 1 \/ ( y - 1 ) e. NN ) ) )
8		eqeq1 2236	. . 3 |- ( x = ( y + 1 ) -> ( x = 1 <-> ( y + 1 ) = 1 ) )
9		oveq1 6020	. . . 4 |- ( x = ( y + 1 ) -> ( x - 1 ) = ( ( y + 1 ) - 1 ) )
10	9	eleq1d 2298	. . 3 |- ( x = ( y + 1 ) -> ( ( x - 1 ) e. NN <-> ( ( y + 1 ) - 1 ) e. NN ) )
11	8, 10	orbi12d 798	. 2 |- ( x = ( y + 1 ) -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( ( y + 1 ) = 1 \/ ( ( y + 1 ) - 1 ) e. NN ) ) )
12		eqeq1 2236	. . 3 |- ( x = A -> ( x = 1 <-> A = 1 ) )
13		oveq1 6020	. . . 4 |- ( x = A -> ( x - 1 ) = ( A - 1 ) )
14	13	eleq1d 2298	. . 3 |- ( x = A -> ( ( x - 1 ) e. NN <-> ( A - 1 ) e. NN ) )
15	12, 14	orbi12d 798	. 2 |- ( x = A -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( A = 1 \/ ( A - 1 ) e. NN ) ) )
16		ax-1cn 8115	. 2 |- 1 e. CC
17		nncn 9141	. . . . . 6 |- ( y e. NN -> y e. CC )
18		pncan 8375	. . . . . 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 2306	. . . 4 |- ( y e. NN -> ( ( y + 1 ) - 1 ) e. NN )
22	21	olcd 739	. . 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 9149	1 |- ( A e. NN -> ( A = 1 \/ ( A - 1 ) e. NN ) )

Syntax hints:   -> wi 4   \/ wo 713   = wceq 1395   e. wcel 2200  (class class class)co 6013   CCcc 8020   1c1 8023   + caddc 8025   - cmin 8340   NNcn 9133

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-sub 8342  df-inn 9134

This theorem is referenced by:  nn1suc  9152  nnsub  9172  nnm1nn0  9433  nn0ge2m1nn  9452