Theorem nn1gt1 9288

Description: A positive integer is either one or greater than one. This is for NN; 0elnn 4746 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 2241	. . 3 |- ( x = 1 -> ( x = 1 <-> 1 = 1 ) )
2		breq2 4118	. . 3 |- ( x = 1 -> ( 1 < x <-> 1 < 1 ) )
3	1, 2	orbi12d 801	. 2 |- ( x = 1 -> ( ( x = 1 \/ 1 < x ) <-> ( 1 = 1 \/ 1 < 1 ) ) )
4		eqeq1 2241	. . 3 |- ( x = y -> ( x = 1 <-> y = 1 ) )
5		breq2 4118	. . 3 |- ( x = y -> ( 1 < x <-> 1 < y ) )
6	4, 5	orbi12d 801	. 2 |- ( x = y -> ( ( x = 1 \/ 1 < x ) <-> ( y = 1 \/ 1 < y ) ) )
7		eqeq1 2241	. . 3 |- ( x = ( y + 1 ) -> ( x = 1 <-> ( y + 1 ) = 1 ) )
8		breq2 4118	. . 3 |- ( x = ( y + 1 ) -> ( 1 < x <-> 1 < ( y + 1 ) ) )
9	7, 8	orbi12d 801	. 2 |- ( x = ( y + 1 ) -> ( ( x = 1 \/ 1 < x ) <-> ( ( y + 1 ) = 1 \/ 1 < ( y + 1 ) ) ) )
10		eqeq1 2241	. . 3 |- ( x = A -> ( x = 1 <-> A = 1 ) )
11		breq2 4118	. . 3 |- ( x = A -> ( 1 < x <-> 1 < A ) )
12	10, 11	orbi12d 801	. 2 |- ( x = A -> ( ( x = 1 \/ 1 < x ) <-> ( A = 1 \/ 1 < A ) ) )
13		eqid 2234	. . 3 |- 1 = 1
14	13	orci 739	. 2 |- ( 1 = 1 \/ 1 < 1 )
15		nngt0 9279	. . . . 5 |- ( y e. NN -> 0 < y )
16		nnre 9261	. . . . . 6 |- ( y e. NN -> y e. RR )
17		1re 8289	. . . . . 6 |- 1 e. RR
18		ltaddpos2 8744	. . . . . 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 742	. . 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 9270	1 |- ( A e. NN -> ( A = 1 \/ 1 < A ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   class class class wbr 4114  (class class class)co 6058   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   NNcn 9254

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-iota 5317  df-fv 5365  df-ov 6061  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-inn 9255

This theorem is referenced by:  nngt1ne1  9289  resqrexlemglsq  11732