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Theorem nn1gt1 8761
Description: A positive integer is either one or greater than one. This is for  NN; 0elnn 4532 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 2146 . . 3  |-  ( x  =  1  ->  (
x  =  1  <->  1  =  1 ) )
2 breq2 3933 . . 3  |-  ( x  =  1  ->  (
1  <  x  <->  1  <  1 ) )
31, 2orbi12d 782 . 2  |-  ( x  =  1  ->  (
( x  =  1  \/  1  <  x
)  <->  ( 1  =  1  \/  1  <  1 ) ) )
4 eqeq1 2146 . . 3  |-  ( x  =  y  ->  (
x  =  1  <->  y  =  1 ) )
5 breq2 3933 . . 3  |-  ( x  =  y  ->  (
1  <  x  <->  1  <  y ) )
64, 5orbi12d 782 . 2  |-  ( x  =  y  ->  (
( x  =  1  \/  1  <  x
)  <->  ( y  =  1  \/  1  < 
y ) ) )
7 eqeq1 2146 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
x  =  1  <->  (
y  +  1 )  =  1 ) )
8 breq2 3933 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
1  <  x  <->  1  <  ( y  +  1 ) ) )
97, 8orbi12d 782 . 2  |-  ( x  =  ( y  +  1 )  ->  (
( x  =  1  \/  1  <  x
)  <->  ( ( y  +  1 )  =  1  \/  1  < 
( y  +  1 ) ) ) )
10 eqeq1 2146 . . 3  |-  ( x  =  A  ->  (
x  =  1  <->  A  =  1 ) )
11 breq2 3933 . . 3  |-  ( x  =  A  ->  (
1  <  x  <->  1  <  A ) )
1210, 11orbi12d 782 . 2  |-  ( x  =  A  ->  (
( x  =  1  \/  1  <  x
)  <->  ( A  =  1  \/  1  < 
A ) ) )
13 eqid 2139 . . 3  |-  1  =  1
1413orci 720 . 2  |-  ( 1  =  1  \/  1  <  1 )
15 nngt0 8752 . . . . 5  |-  ( y  e.  NN  ->  0  <  y )
16 nnre 8734 . . . . . 6  |-  ( y  e.  NN  ->  y  e.  RR )
17 1re 7772 . . . . . 6  |-  1  e.  RR
18 ltaddpos2 8222 . . . . . 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 723 . . 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 8743 1  |-  ( A  e.  NN  ->  ( A  =  1  \/  1  <  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   RRcr 7626   0cc0 7627   1c1 7628    + caddc 7630    < clt 7807   NNcn 8727
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-iota 5088  df-fv 5131  df-ov 5777  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-inn 8728
This theorem is referenced by:  nngt1ne1  8762  resqrexlemglsq  10801
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