Theorem nn1gt1 8072

Description: A positive integer is either one or greater than one. This is for NN; 0elnn 4358 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 2087	. . 3 |- ( x = 1 -> ( x = 1 <-> 1 = 1 ) )
2		breq2 3789	. . 3 |- ( x = 1 -> ( 1 < x <-> 1 < 1 ) )
3	1, 2	orbi12d 739	. 2 |- ( x = 1 -> ( ( x = 1 \/ 1 < x ) <-> ( 1 = 1 \/ 1 < 1 ) ) )
4		eqeq1 2087	. . 3 |- ( x = y -> ( x = 1 <-> y = 1 ) )
5		breq2 3789	. . 3 |- ( x = y -> ( 1 < x <-> 1 < y ) )
6	4, 5	orbi12d 739	. 2 |- ( x = y -> ( ( x = 1 \/ 1 < x ) <-> ( y = 1 \/ 1 < y ) ) )
7		eqeq1 2087	. . 3 |- ( x = ( y + 1 ) -> ( x = 1 <-> ( y + 1 ) = 1 ) )
8		breq2 3789	. . 3 |- ( x = ( y + 1 ) -> ( 1 < x <-> 1 < ( y + 1 ) ) )
9	7, 8	orbi12d 739	. 2 |- ( x = ( y + 1 ) -> ( ( x = 1 \/ 1 < x ) <-> ( ( y + 1 ) = 1 \/ 1 < ( y + 1 ) ) ) )
10		eqeq1 2087	. . 3 |- ( x = A -> ( x = 1 <-> A = 1 ) )
11		breq2 3789	. . 3 |- ( x = A -> ( 1 < x <-> 1 < A ) )
12	10, 11	orbi12d 739	. 2 |- ( x = A -> ( ( x = 1 \/ 1 < x ) <-> ( A = 1 \/ 1 < A ) ) )
13		eqid 2081	. . 3 |- 1 = 1
14	13	orci 682	. 2 |- ( 1 = 1 \/ 1 < 1 )
15		nngt0 8064	. . . . 5 |- ( y e. NN -> 0 < y )
16		nnre 8046	. . . . . 6 |- ( y e. NN -> y e. RR )
17		1re 7118	. . . . . 6 |- 1 e. RR
18		ltaddpos2 7557	. . . . . 6 |- ( ( y e. RR /\ 1 e. RR ) -> ( 0 < y <-> 1 < ( y + 1 ) ) )
19	16, 17, 18	sylancl 404	. . . . 5 |- ( y e. NN -> ( 0 < y <-> 1 < ( y + 1 ) ) )
20	15, 19	mpbid 145	. . . 4 |- ( y e. NN -> 1 < ( y + 1 ) )
21	20	olcd 685	. . 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 8055	1 |- ( A e. NN -> ( A = 1 \/ 1 < A ) )

Syntax hints:   -> wi 4   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   RRcr 6980   0cc0 6981   1c1 6982   + caddc 6984   < clt 7153   NNcn 8039

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-iota 4887  df-fv 4930  df-ov 5535  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-inn 8040

This theorem is referenced by:  nngt1ne1  8073  resqrexlemglsq  9908