Theorem nn1gt1 9144

Description: A positive integer is either one or greater than one. This is for NN; 0elnn 4711 is a similar theorem for om (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)

Assertion

Ref	Expression
nn1gt1	|- ( A e. NN -> ( A = 1 \/ 1 < A ) )

Proof of Theorem nn1gt1

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

Step	Hyp	Ref	Expression
1		eqeq1 2236	. . 3 |- ( x = 1 -> ( x = 1 <-> 1 = 1 ) )
2		breq2 4087	. . 3 |- ( x = 1 -> ( 1 < x <-> 1 < 1 ) )
3	1, 2	orbi12d 798	. 2 |- ( x = 1 -> ( ( x = 1 \/ 1 < x ) <-> ( 1 = 1 \/ 1 < 1 ) ) )
4		eqeq1 2236	. . 3 |- ( x = y -> ( x = 1 <-> y = 1 ) )
5		breq2 4087	. . 3 |- ( x = y -> ( 1 < x <-> 1 < y ) )
6	4, 5	orbi12d 798	. 2 |- ( x = y -> ( ( x = 1 \/ 1 < x ) <-> ( y = 1 \/ 1 < y ) ) )
7		eqeq1 2236	. . 3 |- ( x = ( y + 1 ) -> ( x = 1 <-> ( y + 1 ) = 1 ) )
8		breq2 4087	. . 3 |- ( x = ( y + 1 ) -> ( 1 < x <-> 1 < ( y + 1 ) ) )
9	7, 8	orbi12d 798	. 2 |- ( x = ( y + 1 ) -> ( ( x = 1 \/ 1 < x ) <-> ( ( y + 1 ) = 1 \/ 1 < ( y + 1 ) ) ) )
10		eqeq1 2236	. . 3 |- ( x = A -> ( x = 1 <-> A = 1 ) )
11		breq2 4087	. . 3 |- ( x = A -> ( 1 < x <-> 1 < A ) )
12	10, 11	orbi12d 798	. 2 |- ( x = A -> ( ( x = 1 \/ 1 < x ) <-> ( A = 1 \/ 1 < A ) ) )
13		eqid 2229	. . 3 |- 1 = 1
14	13	orci 736	. 2 |- ( 1 = 1 \/ 1 < 1 )
15		nngt0 9135	. . . . 5 |- ( y e. NN -> 0 < y )
16		nnre 9117	. . . . . 6 |- ( y e. NN -> y e. RR )
17		1re 8145	. . . . . 6 |- 1 e. RR
18		ltaddpos2 8600	. . . . . 6 |- ( ( y e. RR /\ 1 e. RR ) -> ( 0 < y <-> 1 < ( y + 1 ) ) )
19	16, 17, 18	sylancl 413	. . . . 5 |- ( y e. NN -> ( 0 < y <-> 1 < ( y + 1 ) ) )
20	15, 19	mpbid 147	. . . 4 |- ( y e. NN -> 1 < ( y + 1 ) )
21	20	olcd 739	. . 3 |- ( y e. NN -> ( ( y + 1 ) = 1 \/ 1 < ( y + 1 ) ) )
22	21	a1d 22	. 2 |- ( y e. NN -> ( ( y = 1 \/ 1 < y ) -> ( ( y + 1 ) = 1 \/ 1 < ( y + 1 ) ) ) )
23	3, 6, 9, 12, 14, 22	nnind 9126	1 |- ( A e. NN -> ( A = 1 \/ 1 < A ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   RRcr 7998   0cc0 7999   1c1 8000   + caddc 8002   < clt 8181   NNcn 9110

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-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

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-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-iota 5278  df-fv 5326  df-ov 6004  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-inn 9111

This theorem is referenced by:  nngt1ne1  9145  resqrexlemglsq  11533