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Theorem nlt1pi 8530
Description: No positive integer is less than one. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
Assertion
Ref Expression
nlt1pi  |-  -.  A  <N  1o

Proof of Theorem nlt1pi
StepHypRef Expression
1 elni 8500 . . . 4  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
21simprbi 450 . . 3  |-  ( A  e.  N.  ->  A  =/=  (/) )
3 noel 3459 . . . . . 6  |-  -.  A  e.  (/)
4 1pi 8507 . . . . . . . . . 10  |-  1o  e.  N.
5 ltpiord 8511 . . . . . . . . . 10  |-  ( ( A  e.  N.  /\  1o  e.  N. )  -> 
( A  <N  1o  <->  A  e.  1o ) )
64, 5mpan2 652 . . . . . . . . 9  |-  ( A  e.  N.  ->  ( A  <N  1o  <->  A  e.  1o ) )
7 df-1o 6479 . . . . . . . . . . 11  |-  1o  =  suc  (/)
87eleq2i 2347 . . . . . . . . . 10  |-  ( A  e.  1o  <->  A  e.  suc  (/) )
9 elsucg 4459 . . . . . . . . . 10  |-  ( A  e.  N.  ->  ( A  e.  suc  (/)  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
108, 9syl5bb 248 . . . . . . . . 9  |-  ( A  e.  N.  ->  ( A  e.  1o  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
116, 10bitrd 244 . . . . . . . 8  |-  ( A  e.  N.  ->  ( A  <N  1o  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
1211biimpa 470 . . . . . . 7  |-  ( ( A  e.  N.  /\  A  <N  1o )  -> 
( A  e.  (/)  \/  A  =  (/) ) )
1312ord 366 . . . . . 6  |-  ( ( A  e.  N.  /\  A  <N  1o )  -> 
( -.  A  e.  (/)  ->  A  =  (/) ) )
143, 13mpi 16 . . . . 5  |-  ( ( A  e.  N.  /\  A  <N  1o )  ->  A  =  (/) )
1514ex 423 . . . 4  |-  ( A  e.  N.  ->  ( A  <N  1o  ->  A  =  (/) ) )
1615necon3ad 2482 . . 3  |-  ( A  e.  N.  ->  ( A  =/=  (/)  ->  -.  A  <N  1o ) )
172, 16mpd 14 . 2  |-  ( A  e.  N.  ->  -.  A  <N  1o )
18 ltrelpi 8513 . . . . 5  |-  <N  C_  ( N.  X.  N. )
1918brel 4737 . . . 4  |-  ( A 
<N  1o  ->  ( A  e.  N.  /\  1o  e.  N. ) )
2019simpld 445 . . 3  |-  ( A 
<N  1o  ->  A  e.  N. )
2120con3i 127 . 2  |-  ( -.  A  e.  N.  ->  -.  A  <N  1o )
2217, 21pm2.61i 156 1  |-  -.  A  <N  1o
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   class class class wbr 4023   suc csuc 4394   omcom 4656   1oc1o 6472   N.cnpi 8466    <N clti 8469
This theorem is referenced by:  indpi  8531  pinq  8551  archnq  8604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-1o 6479  df-ni 8496  df-lti 8499
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