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Theorem nn0ge2m1nnALT 9614
Description: Alternate proof of nn0ge2m1nn 9232: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. This version is proved using eluz2 9530, a theorem for upper sets of integers, which are defined later than the positive and nonnegative integers. This proof is, however, much shorter than the proof of nn0ge2m1nn 9232. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
nn0ge2m1nnALT  |-  ( ( N  e.  NN0  /\  2  <_  N )  -> 
( N  -  1 )  e.  NN )

Proof of Theorem nn0ge2m1nnALT
StepHypRef Expression
1 2z 9277 . . . 4  |-  2  e.  ZZ
21a1i 9 . . 3  |-  ( ( N  e.  NN0  /\  2  <_  N )  -> 
2  e.  ZZ )
3 nn0z 9269 . . . 4  |-  ( N  e.  NN0  ->  N  e.  ZZ )
43adantr 276 . . 3  |-  ( ( N  e.  NN0  /\  2  <_  N )  ->  N  e.  ZZ )
5 simpr 110 . . 3  |-  ( ( N  e.  NN0  /\  2  <_  N )  -> 
2  <_  N )
6 eluz2 9530 . . 3  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  N  e.  ZZ  /\  2  <_  N ) )
72, 4, 5, 6syl3anbrc 1181 . 2  |-  ( ( N  e.  NN0  /\  2  <_  N )  ->  N  e.  ( ZZ>= ` 
2 ) )
8 uz2m1nn 9601 . 2  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  1 )  e.  NN )
97, 8syl 14 1  |-  ( ( N  e.  NN0  /\  2  <_  N )  -> 
( N  -  1 )  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2148   class class class wbr 4002   ` cfv 5215  (class class class)co 5872   1c1 7809    <_ cle 7989    - cmin 8124   NNcn 8915   2c2 8966   NN0cn0 9172   ZZcz 9249   ZZ>=cuz 9524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-distr 7912  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-cnre 7919  ax-pre-ltirr 7920  ax-pre-ltwlin 7921  ax-pre-lttrn 7922  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-xr 7992  df-ltxr 7993  df-le 7994  df-sub 8126  df-neg 8127  df-inn 8916  df-2 8974  df-n0 9173  df-z 9250  df-uz 9525
This theorem is referenced by: (None)
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