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Theorem nn1gt1 8983
Description: A positive integer is either one or greater than one. This is for  NN; 0elnn 4636 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.
StepHypRef Expression
1 eqeq1 2196 . . 3  |-  ( x  =  1  ->  (
x  =  1  <->  1  =  1 ) )
2 breq2 4022 . . 3  |-  ( x  =  1  ->  (
1  <  x  <->  1  <  1 ) )
31, 2orbi12d 794 . 2  |-  ( x  =  1  ->  (
( x  =  1  \/  1  <  x
)  <->  ( 1  =  1  \/  1  <  1 ) ) )
4 eqeq1 2196 . . 3  |-  ( x  =  y  ->  (
x  =  1  <->  y  =  1 ) )
5 breq2 4022 . . 3  |-  ( x  =  y  ->  (
1  <  x  <->  1  <  y ) )
64, 5orbi12d 794 . 2  |-  ( x  =  y  ->  (
( x  =  1  \/  1  <  x
)  <->  ( y  =  1  \/  1  < 
y ) ) )
7 eqeq1 2196 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
x  =  1  <->  (
y  +  1 )  =  1 ) )
8 breq2 4022 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
1  <  x  <->  1  <  ( y  +  1 ) ) )
97, 8orbi12d 794 . 2  |-  ( x  =  ( y  +  1 )  ->  (
( x  =  1  \/  1  <  x
)  <->  ( ( y  +  1 )  =  1  \/  1  < 
( y  +  1 ) ) ) )
10 eqeq1 2196 . . 3  |-  ( x  =  A  ->  (
x  =  1  <->  A  =  1 ) )
11 breq2 4022 . . 3  |-  ( x  =  A  ->  (
1  <  x  <->  1  <  A ) )
1210, 11orbi12d 794 . 2  |-  ( x  =  A  ->  (
( x  =  1  \/  1  <  x
)  <->  ( A  =  1  \/  1  < 
A ) ) )
13 eqid 2189 . . 3  |-  1  =  1
1413orci 732 . 2  |-  ( 1  =  1  \/  1  <  1 )
15 nngt0 8974 . . . . 5  |-  ( y  e.  NN  ->  0  <  y )
16 nnre 8956 . . . . . 6  |-  ( y  e.  NN  ->  y  e.  RR )
17 1re 7986 . . . . . 6  |-  1  e.  RR
18 ltaddpos2 8440 . . . . . 6  |-  ( ( y  e.  RR  /\  1  e.  RR )  ->  ( 0  <  y  <->  1  <  ( y  +  1 ) ) )
1916, 17, 18sylancl 413 . . . . 5  |-  ( y  e.  NN  ->  (
0  <  y  <->  1  <  ( y  +  1 ) ) )
2015, 19mpbid 147 . . . 4  |-  ( y  e.  NN  ->  1  <  ( y  +  1 ) )
2120olcd 735 . . 3  |-  ( y  e.  NN  ->  (
( y  +  1 )  =  1  \/  1  <  ( y  +  1 ) ) )
2221a1d 22 . 2  |-  ( y  e.  NN  ->  (
( y  =  1  \/  1  <  y
)  ->  ( (
y  +  1 )  =  1  \/  1  <  ( y  +  1 ) ) ) )
233, 6, 9, 12, 14, 22nnind 8965 1  |-  ( A  e.  NN  ->  ( A  =  1  \/  1  <  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5896   RRcr 7840   0cc0 7841   1c1 7842    + caddc 7844    < clt 8022   NNcn 8949
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-i2m1 7946  ax-0lt1 7947  ax-0id 7949  ax-rnegex 7950  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-ltadd 7957
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-iota 5196  df-fv 5243  df-ov 5899  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-inn 8950
This theorem is referenced by:  nngt1ne1  8984  resqrexlemglsq  11063
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