Theorem nn1m1nn 8939

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 712	. . 3 |- ( x = 1 -> ( x = 1 \/ ( x - 1 ) e. NN ) )
2		1cnd 7975	. . 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 2184	. . 3 |- ( x = y -> ( x = 1 <-> y = 1 ) )
5		oveq1 5884	. . . 4 |- ( x = y -> ( x - 1 ) = ( y - 1 ) )
6	5	eleq1d 2246	. . 3 |- ( x = y -> ( ( x - 1 ) e. NN <-> ( y - 1 ) e. NN ) )
7	4, 6	orbi12d 793	. 2 |- ( x = y -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( y = 1 \/ ( y - 1 ) e. NN ) ) )
8		eqeq1 2184	. . 3 |- ( x = ( y + 1 ) -> ( x = 1 <-> ( y + 1 ) = 1 ) )
9		oveq1 5884	. . . 4 |- ( x = ( y + 1 ) -> ( x - 1 ) = ( ( y + 1 ) - 1 ) )
10	9	eleq1d 2246	. . 3 |- ( x = ( y + 1 ) -> ( ( x - 1 ) e. NN <-> ( ( y + 1 ) - 1 ) e. NN ) )
11	8, 10	orbi12d 793	. 2 |- ( x = ( y + 1 ) -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( ( y + 1 ) = 1 \/ ( ( y + 1 ) - 1 ) e. NN ) ) )
12		eqeq1 2184	. . 3 |- ( x = A -> ( x = 1 <-> A = 1 ) )
13		oveq1 5884	. . . 4 |- ( x = A -> ( x - 1 ) = ( A - 1 ) )
14	13	eleq1d 2246	. . 3 |- ( x = A -> ( ( x - 1 ) e. NN <-> ( A - 1 ) e. NN ) )
15	12, 14	orbi12d 793	. 2 |- ( x = A -> ( ( x = 1 \/ ( x - 1 ) e. NN ) <-> ( A = 1 \/ ( A - 1 ) e. NN ) ) )
16		ax-1cn 7906	. 2 |- 1 e. CC
17		nncn 8929	. . . . . 6 |- ( y e. NN -> y e. CC )
18		pncan 8165	. . . . . 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 2254	. . . 4 |- ( y e. NN -> ( ( y + 1 ) - 1 ) e. NN )
22	21	olcd 734	. . 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 8937	1 |- ( A e. NN -> ( A = 1 \/ ( A - 1 ) e. NN ) )

Syntax hints:   -> wi 4   \/ wo 708   = wceq 1353   e. wcel 2148  (class class class)co 5877   CCcc 7811   1c1 7814   + caddc 7816   - cmin 8130   NNcn 8921

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-sub 8132  df-inn 8922

This theorem is referenced by:  nn1suc  8940  nnsub  8960  nnm1nn0  9219  nn0ge2m1nn  9238