Theorem nn1m1nn 8851

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 702	. . 3 |- ( x = 1 -> ( x = 1 \/ ( x - 1 ) e. NN ) )
2		1cnd 7894	. . 3 |- ( x = 1 -> 1 e. CC )
3	1, 2	2thd 174	. 2 |- ( x = 1 -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> 1 e. CC ) )
4		eqeq1 2164	. . 3 |- ( x = y -> ( x = 1 <-> y = 1 ) )
5		oveq1 5831	. . . 4 |- ( x = y -> ( x - 1 ) = ( y - 1 ) )
6	5	eleq1d 2226	. . 3 |- ( x = y -> ( ( x - 1 ) e. NN <-> ( y - 1 ) e. NN ) )
7	4, 6	orbi12d 783	. 2 |- ( x = y -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( y = 1 \/ ( y - 1 ) e. NN ) ) )
8		eqeq1 2164	. . 3 |- ( x = ( y + 1 ) -> ( x = 1 <-> ( y + 1 ) = 1 ) )
9		oveq1 5831	. . . 4 |- ( x = ( y + 1 ) -> ( x - 1 ) = ( ( y + 1 ) - 1 ) )
10	9	eleq1d 2226	. . 3 |- ( x = ( y + 1 ) -> ( ( x - 1 ) e. NN <-> ( ( y + 1 ) - 1 ) e. NN ) )
11	8, 10	orbi12d 783	. 2 |- ( x = ( y + 1 ) -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( ( y + 1 ) = 1 \/ ( ( y + 1 ) - 1 ) e. NN ) ) )
12		eqeq1 2164	. . 3 |- ( x = A -> ( x = 1 <-> A = 1 ) )
13		oveq1 5831	. . . 4 |- ( x = A -> ( x - 1 ) = ( A - 1 ) )
14	13	eleq1d 2226	. . 3 |- ( x = A -> ( ( x - 1 ) e. NN <-> ( A - 1 ) e. NN ) )
15	12, 14	orbi12d 783	. 2 |- ( x = A -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( A = 1 \/ ( A - 1 ) e. NN ) ) )
16		ax-1cn 7825	. 2 |- 1 e. CC
17		nncn 8841	. . . . . 6 |- ( y e. NN -> y e. CC )
18		pncan 8081	. . . . . 6 |- ( ( y e. CC /\ 1 e. CC ) -> ( ( y + 1 ) - 1 ) = y )
19	17, 16, 18	sylancl 410	. . . . 5 |- ( y e. NN -> ( ( y + 1 ) - 1 ) = y )
20		id 19	. . . . 5 |- ( y e. NN -> y e. NN )
21	19, 20	eqeltrd 2234	. . . 4 |- ( y e. NN -> ( ( y + 1 ) - 1 ) e. NN )
22	21	olcd 724	. . 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 8849	1 |- ( A e. NN -> ( A = 1 \/ ( A - 1 ) e. NN ) )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1335   e. wcel 2128  (class class class)co 5824   CCcc 7730   1c1 7733   + caddc 7735   - cmin 8046   NNcn 8833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-sub 8048  df-inn 8834

This theorem is referenced by:  nn1suc  8852  nnsub  8872  nnm1nn0  9131  nn0ge2m1nn  9150