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Theorem nn1m1nn 8176
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 666 . . 3  |-  ( x  =  1  ->  (
x  =  1  \/  ( x  -  1 )  e.  NN ) )
2 1cnd 7249 . . 3  |-  ( x  =  1  ->  1  e.  CC )
31, 22thd 173 . 2  |-  ( x  =  1  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  1  e.  CC ) )
4 eqeq1 2089 . . 3  |-  ( x  =  y  ->  (
x  =  1  <->  y  =  1 ) )
5 oveq1 5570 . . . 4  |-  ( x  =  y  ->  (
x  -  1 )  =  ( y  - 
1 ) )
65eleq1d 2151 . . 3  |-  ( x  =  y  ->  (
( x  -  1 )  e.  NN  <->  ( y  -  1 )  e.  NN ) )
74, 6orbi12d 740 . 2  |-  ( x  =  y  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  ( y  =  1  \/  ( y  - 
1 )  e.  NN ) ) )
8 eqeq1 2089 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
x  =  1  <->  (
y  +  1 )  =  1 ) )
9 oveq1 5570 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  -  1 )  =  ( ( y  +  1 )  - 
1 ) )
109eleq1d 2151 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( x  -  1 )  e.  NN  <->  ( (
y  +  1 )  -  1 )  e.  NN ) )
118, 10orbi12d 740 . 2  |-  ( x  =  ( y  +  1 )  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  ( ( y  +  1 )  =  1  \/  ( ( y  +  1 )  - 
1 )  e.  NN ) ) )
12 eqeq1 2089 . . 3  |-  ( x  =  A  ->  (
x  =  1  <->  A  =  1 ) )
13 oveq1 5570 . . . 4  |-  ( x  =  A  ->  (
x  -  1 )  =  ( A  - 
1 ) )
1413eleq1d 2151 . . 3  |-  ( x  =  A  ->  (
( x  -  1 )  e.  NN  <->  ( A  -  1 )  e.  NN ) )
1512, 14orbi12d 740 . 2  |-  ( x  =  A  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  ( A  =  1  \/  ( A  - 
1 )  e.  NN ) ) )
16 ax-1cn 7183 . 2  |-  1  e.  CC
17 nncn 8166 . . . . . 6  |-  ( y  e.  NN  ->  y  e.  CC )
18 pncan 7433 . . . . . 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 2159 . . . 4  |-  ( y  e.  NN  ->  (
( y  +  1 )  -  1 )  e.  NN )
2221olcd 686 . . 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 8174 1  |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1
)  e.  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662    = wceq 1285    e. wcel 1434  (class class class)co 5563   CCcc 7093   1c1 7096    + caddc 7098    - cmin 7398   NNcn 8158
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-sub 7400  df-inn 8159
This theorem is referenced by:  nn1suc  8177  nnsub  8196  nnm1nn0  8448  nn0ge2m1nn  8467
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