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Theorem nn1gt1 8367
Description: A positive integer is either one or greater than one. This is for  NN; 0elnn 4398 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 2091 . . 3  |-  ( x  =  1  ->  (
x  =  1  <->  1  =  1 ) )
2 breq2 3818 . . 3  |-  ( x  =  1  ->  (
1  <  x  <->  1  <  1 ) )
31, 2orbi12d 740 . 2  |-  ( x  =  1  ->  (
( x  =  1  \/  1  <  x
)  <->  ( 1  =  1  \/  1  <  1 ) ) )
4 eqeq1 2091 . . 3  |-  ( x  =  y  ->  (
x  =  1  <->  y  =  1 ) )
5 breq2 3818 . . 3  |-  ( x  =  y  ->  (
1  <  x  <->  1  <  y ) )
64, 5orbi12d 740 . 2  |-  ( x  =  y  ->  (
( x  =  1  \/  1  <  x
)  <->  ( y  =  1  \/  1  < 
y ) ) )
7 eqeq1 2091 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
x  =  1  <->  (
y  +  1 )  =  1 ) )
8 breq2 3818 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
1  <  x  <->  1  <  ( y  +  1 ) ) )
97, 8orbi12d 740 . 2  |-  ( x  =  ( y  +  1 )  ->  (
( x  =  1  \/  1  <  x
)  <->  ( ( y  +  1 )  =  1  \/  1  < 
( y  +  1 ) ) ) )
10 eqeq1 2091 . . 3  |-  ( x  =  A  ->  (
x  =  1  <->  A  =  1 ) )
11 breq2 3818 . . 3  |-  ( x  =  A  ->  (
1  <  x  <->  1  <  A ) )
1210, 11orbi12d 740 . 2  |-  ( x  =  A  ->  (
( x  =  1  \/  1  <  x
)  <->  ( A  =  1  \/  1  < 
A ) ) )
13 eqid 2085 . . 3  |-  1  =  1
1413orci 683 . 2  |-  ( 1  =  1  \/  1  <  1 )
15 nngt0 8359 . . . . 5  |-  ( y  e.  NN  ->  0  <  y )
16 nnre 8341 . . . . . 6  |-  ( y  e.  NN  ->  y  e.  RR )
17 1re 7408 . . . . . 6  |-  1  e.  RR
18 ltaddpos2 7852 . . . . . 6  |-  ( ( y  e.  RR  /\  1  e.  RR )  ->  ( 0  <  y  <->  1  <  ( y  +  1 ) ) )
1916, 17, 18sylancl 404 . . . . 5  |-  ( y  e.  NN  ->  (
0  <  y  <->  1  <  ( y  +  1 ) ) )
2015, 19mpbid 145 . . . 4  |-  ( y  e.  NN  ->  1  <  ( y  +  1 ) )
2120olcd 686 . . 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 8350 1  |-  ( A  e.  NN  ->  ( A  =  1  \/  1  <  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    \/ wo 662    = wceq 1287    e. wcel 1436   class class class wbr 3814  (class class class)co 5594   RRcr 7270   0cc0 7271   1c1 7272    + caddc 7274    < clt 7443   NNcn 8334
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-addcom 7366  ax-addass 7368  ax-i2m1 7371  ax-0lt1 7372  ax-0id 7374  ax-rnegex 7375  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-ltadd 7382
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2616  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-br 3815  df-opab 3869  df-xp 4410  df-cnv 4412  df-iota 4937  df-fv 4980  df-ov 5597  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-inn 8335
This theorem is referenced by:  nngt1ne1  8368  resqrexlemglsq  10296
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