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Theorem nn1m1nn 8896
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 707 . . 3  |-  ( x  =  1  ->  (
x  =  1  \/  ( x  -  1 )  e.  NN ) )
2 1cnd 7936 . . 3  |-  ( x  =  1  ->  1  e.  CC )
31, 22thd 174 . 2  |-  ( x  =  1  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  1  e.  CC ) )
4 eqeq1 2177 . . 3  |-  ( x  =  y  ->  (
x  =  1  <->  y  =  1 ) )
5 oveq1 5860 . . . 4  |-  ( x  =  y  ->  (
x  -  1 )  =  ( y  - 
1 ) )
65eleq1d 2239 . . 3  |-  ( x  =  y  ->  (
( x  -  1 )  e.  NN  <->  ( y  -  1 )  e.  NN ) )
74, 6orbi12d 788 . 2  |-  ( x  =  y  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  ( y  =  1  \/  ( y  - 
1 )  e.  NN ) ) )
8 eqeq1 2177 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
x  =  1  <->  (
y  +  1 )  =  1 ) )
9 oveq1 5860 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  -  1 )  =  ( ( y  +  1 )  - 
1 ) )
109eleq1d 2239 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( x  -  1 )  e.  NN  <->  ( (
y  +  1 )  -  1 )  e.  NN ) )
118, 10orbi12d 788 . 2  |-  ( x  =  ( y  +  1 )  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  ( ( y  +  1 )  =  1  \/  ( ( y  +  1 )  - 
1 )  e.  NN ) ) )
12 eqeq1 2177 . . 3  |-  ( x  =  A  ->  (
x  =  1  <->  A  =  1 ) )
13 oveq1 5860 . . . 4  |-  ( x  =  A  ->  (
x  -  1 )  =  ( A  - 
1 ) )
1413eleq1d 2239 . . 3  |-  ( x  =  A  ->  (
( x  -  1 )  e.  NN  <->  ( A  -  1 )  e.  NN ) )
1512, 14orbi12d 788 . 2  |-  ( x  =  A  ->  (
( x  =  1  \/  ( x  - 
1 )  e.  NN ) 
<->  ( A  =  1  \/  ( A  - 
1 )  e.  NN ) ) )
16 ax-1cn 7867 . 2  |-  1  e.  CC
17 nncn 8886 . . . . . 6  |-  ( y  e.  NN  ->  y  e.  CC )
18 pncan 8125 . . . . . 6  |-  ( ( y  e.  CC  /\  1  e.  CC )  ->  ( ( y  +  1 )  -  1 )  =  y )
1917, 16, 18sylancl 411 . . . . 5  |-  ( y  e.  NN  ->  (
( y  +  1 )  -  1 )  =  y )
20 id 19 . . . . 5  |-  ( y  e.  NN  ->  y  e.  NN )
2119, 20eqeltrd 2247 . . . 4  |-  ( y  e.  NN  ->  (
( y  +  1 )  -  1 )  e.  NN )
2221olcd 729 . . 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 8894 1  |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1
)  e.  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 703    = wceq 1348    e. wcel 2141  (class class class)co 5853   CCcc 7772   1c1 7775    + caddc 7777    - cmin 8090   NNcn 8878
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092  df-inn 8879
This theorem is referenced by:  nn1suc  8897  nnsub  8917  nnm1nn0  9176  nn0ge2m1nn  9195
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