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Theorem nn1gt1 8722
Description: A positive integer is either one or greater than one. This is for  NN; 0elnn 4502 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 2124 . . 3  |-  ( x  =  1  ->  (
x  =  1  <->  1  =  1 ) )
2 breq2 3903 . . 3  |-  ( x  =  1  ->  (
1  <  x  <->  1  <  1 ) )
31, 2orbi12d 767 . 2  |-  ( x  =  1  ->  (
( x  =  1  \/  1  <  x
)  <->  ( 1  =  1  \/  1  <  1 ) ) )
4 eqeq1 2124 . . 3  |-  ( x  =  y  ->  (
x  =  1  <->  y  =  1 ) )
5 breq2 3903 . . 3  |-  ( x  =  y  ->  (
1  <  x  <->  1  <  y ) )
64, 5orbi12d 767 . 2  |-  ( x  =  y  ->  (
( x  =  1  \/  1  <  x
)  <->  ( y  =  1  \/  1  < 
y ) ) )
7 eqeq1 2124 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
x  =  1  <->  (
y  +  1 )  =  1 ) )
8 breq2 3903 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
1  <  x  <->  1  <  ( y  +  1 ) ) )
97, 8orbi12d 767 . 2  |-  ( x  =  ( y  +  1 )  ->  (
( x  =  1  \/  1  <  x
)  <->  ( ( y  +  1 )  =  1  \/  1  < 
( y  +  1 ) ) ) )
10 eqeq1 2124 . . 3  |-  ( x  =  A  ->  (
x  =  1  <->  A  =  1 ) )
11 breq2 3903 . . 3  |-  ( x  =  A  ->  (
1  <  x  <->  1  <  A ) )
1210, 11orbi12d 767 . 2  |-  ( x  =  A  ->  (
( x  =  1  \/  1  <  x
)  <->  ( A  =  1  \/  1  < 
A ) ) )
13 eqid 2117 . . 3  |-  1  =  1
1413orci 705 . 2  |-  ( 1  =  1  \/  1  <  1 )
15 nngt0 8713 . . . . 5  |-  ( y  e.  NN  ->  0  <  y )
16 nnre 8695 . . . . . 6  |-  ( y  e.  NN  ->  y  e.  RR )
17 1re 7733 . . . . . 6  |-  1  e.  RR
18 ltaddpos2 8183 . . . . . 6  |-  ( ( y  e.  RR  /\  1  e.  RR )  ->  ( 0  <  y  <->  1  <  ( y  +  1 ) ) )
1916, 17, 18sylancl 409 . . . . 5  |-  ( y  e.  NN  ->  (
0  <  y  <->  1  <  ( y  +  1 ) ) )
2015, 19mpbid 146 . . . 4  |-  ( y  e.  NN  ->  1  <  ( y  +  1 ) )
2120olcd 708 . . 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 8704 1  |-  ( A  e.  NN  ->  ( A  =  1  \/  1  <  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 682    = wceq 1316    e. wcel 1465   class class class wbr 3899  (class class class)co 5742   RRcr 7587   0cc0 7588   1c1 7589    + caddc 7591    < clt 7768   NNcn 8688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-iota 5058  df-fv 5101  df-ov 5745  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-inn 8689
This theorem is referenced by:  nngt1ne1  8723  resqrexlemglsq  10762
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