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Theorem nn1m1nn 8706
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 686 . . 3  |-  ( x  =  1  ->  (
x  =  1  \/  ( x  -  1 )  e.  NN ) )
2 1cnd 7750 . . 3  |-  ( x  =  1  ->  1  e.  CC )
31, 22thd 174 . 2  |-  ( x  =  1  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  1  e.  CC ) )
4 eqeq1 2124 . . 3  |-  ( x  =  y  ->  (
x  =  1  <->  y  =  1 ) )
5 oveq1 5749 . . . 4  |-  ( x  =  y  ->  (
x  -  1 )  =  ( y  - 
1 ) )
65eleq1d 2186 . . 3  |-  ( x  =  y  ->  (
( x  -  1 )  e.  NN  <->  ( y  -  1 )  e.  NN ) )
74, 6orbi12d 767 . 2  |-  ( x  =  y  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  ( y  =  1  \/  ( y  - 
1 )  e.  NN ) ) )
8 eqeq1 2124 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
x  =  1  <->  (
y  +  1 )  =  1 ) )
9 oveq1 5749 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  -  1 )  =  ( ( y  +  1 )  - 
1 ) )
109eleq1d 2186 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( x  -  1 )  e.  NN  <->  ( (
y  +  1 )  -  1 )  e.  NN ) )
118, 10orbi12d 767 . 2  |-  ( x  =  ( y  +  1 )  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  ( ( y  +  1 )  =  1  \/  ( ( y  +  1 )  - 
1 )  e.  NN ) ) )
12 eqeq1 2124 . . 3  |-  ( x  =  A  ->  (
x  =  1  <->  A  =  1 ) )
13 oveq1 5749 . . . 4  |-  ( x  =  A  ->  (
x  -  1 )  =  ( A  - 
1 ) )
1413eleq1d 2186 . . 3  |-  ( x  =  A  ->  (
( x  -  1 )  e.  NN  <->  ( A  -  1 )  e.  NN ) )
1512, 14orbi12d 767 . 2  |-  ( x  =  A  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  ( A  =  1  \/  ( A  - 
1 )  e.  NN ) ) )
16 ax-1cn 7681 . 2  |-  1  e.  CC
17 nncn 8696 . . . . . 6  |-  ( y  e.  NN  ->  y  e.  CC )
18 pncan 7936 . . . . . 6  |-  ( ( y  e.  CC  /\  1  e.  CC )  ->  ( ( y  +  1 )  -  1 )  =  y )
1917, 16, 18sylancl 409 . . . . 5  |-  ( y  e.  NN  ->  (
( y  +  1 )  -  1 )  =  y )
20 id 19 . . . . 5  |-  ( y  e.  NN  ->  y  e.  NN )
2119, 20eqeltrd 2194 . . . 4  |-  ( y  e.  NN  ->  (
( y  +  1 )  -  1 )  e.  NN )
2221olcd 708 . . 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 8704 1  |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1
)  e.  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 682    = wceq 1316    e. wcel 1465  (class class class)co 5742   CCcc 7586   1c1 7589    + caddc 7591    - cmin 7901   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-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-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-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
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-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  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-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-sub 7903  df-inn 8689
This theorem is referenced by:  nn1suc  8707  nnsub  8727  nnm1nn0  8986  nn0ge2m1nn  9005
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