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Theorem nn1m1nn 8438
Description: Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
Assertion
Ref Expression
nn1m1nn  |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1
)  e.  NN ) )

Proof of Theorem nn1m1nn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 orc 668 . . 3  |-  ( x  =  1  ->  (
x  =  1  \/  ( x  -  1 )  e.  NN ) )
2 1cnd 7502 . . 3  |-  ( x  =  1  ->  1  e.  CC )
31, 22thd 173 . 2  |-  ( x  =  1  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  1  e.  CC ) )
4 eqeq1 2094 . . 3  |-  ( x  =  y  ->  (
x  =  1  <->  y  =  1 ) )
5 oveq1 5659 . . . 4  |-  ( x  =  y  ->  (
x  -  1 )  =  ( y  - 
1 ) )
65eleq1d 2156 . . 3  |-  ( x  =  y  ->  (
( x  -  1 )  e.  NN  <->  ( y  -  1 )  e.  NN ) )
74, 6orbi12d 742 . 2  |-  ( x  =  y  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  ( y  =  1  \/  ( y  - 
1 )  e.  NN ) ) )
8 eqeq1 2094 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
x  =  1  <->  (
y  +  1 )  =  1 ) )
9 oveq1 5659 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  -  1 )  =  ( ( y  +  1 )  - 
1 ) )
109eleq1d 2156 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( x  -  1 )  e.  NN  <->  ( (
y  +  1 )  -  1 )  e.  NN ) )
118, 10orbi12d 742 . 2  |-  ( x  =  ( y  +  1 )  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  ( ( y  +  1 )  =  1  \/  ( ( y  +  1 )  - 
1 )  e.  NN ) ) )
12 eqeq1 2094 . . 3  |-  ( x  =  A  ->  (
x  =  1  <->  A  =  1 ) )
13 oveq1 5659 . . . 4  |-  ( x  =  A  ->  (
x  -  1 )  =  ( A  - 
1 ) )
1413eleq1d 2156 . . 3  |-  ( x  =  A  ->  (
( x  -  1 )  e.  NN  <->  ( A  -  1 )  e.  NN ) )
1512, 14orbi12d 742 . 2  |-  ( x  =  A  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  ( A  =  1  \/  ( A  - 
1 )  e.  NN ) ) )
16 ax-1cn 7436 . 2  |-  1  e.  CC
17 nncn 8428 . . . . . 6  |-  ( y  e.  NN  ->  y  e.  CC )
18 pncan 7686 . . . . . 6  |-  ( ( y  e.  CC  /\  1  e.  CC )  ->  ( ( y  +  1 )  -  1 )  =  y )
1917, 16, 18sylancl 404 . . . . 5  |-  ( y  e.  NN  ->  (
( y  +  1 )  -  1 )  =  y )
20 id 19 . . . . 5  |-  ( y  e.  NN  ->  y  e.  NN )
2119, 20eqeltrd 2164 . . . 4  |-  ( y  e.  NN  ->  (
( y  +  1 )  -  1 )  e.  NN )
2221olcd 688 . . 3  |-  ( y  e.  NN  ->  (
( y  +  1 )  =  1  \/  ( ( y  +  1 )  -  1 )  e.  NN ) )
2322a1d 22 . 2  |-  ( y  e.  NN  ->  (
( y  =  1  \/  ( y  - 
1 )  e.  NN )  ->  ( ( y  +  1 )  =  1  \/  ( ( y  +  1 )  -  1 )  e.  NN ) ) )
243, 7, 11, 15, 16, 23nnind 8436 1  |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1
)  e.  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 664    = wceq 1289    e. wcel 1438  (class class class)co 5652   CCcc 7346   1c1 7349    + caddc 7351    - cmin 7651   NNcn 8420
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-sub 7653  df-inn 8421
This theorem is referenced by:  nn1suc  8439  nnsub  8459  nnm1nn0  8712  nn0ge2m1nn  8731
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