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Theorem nlt1pi 8772
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 8742 . . . 4  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
21simprbi 451 . . 3  |-  ( A  e.  N.  ->  A  =/=  (/) )
3 noel 3624 . . . . . 6  |-  -.  A  e.  (/)
4 1pi 8749 . . . . . . . . . 10  |-  1o  e.  N.
5 ltpiord 8753 . . . . . . . . . 10  |-  ( ( A  e.  N.  /\  1o  e.  N. )  -> 
( A  <N  1o  <->  A  e.  1o ) )
64, 5mpan2 653 . . . . . . . . 9  |-  ( A  e.  N.  ->  ( A  <N  1o  <->  A  e.  1o ) )
7 df-1o 6715 . . . . . . . . . . 11  |-  1o  =  suc  (/)
87eleq2i 2499 . . . . . . . . . 10  |-  ( A  e.  1o  <->  A  e.  suc  (/) )
9 elsucg 4640 . . . . . . . . . 10  |-  ( A  e.  N.  ->  ( A  e.  suc  (/)  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
108, 9syl5bb 249 . . . . . . . . 9  |-  ( A  e.  N.  ->  ( A  e.  1o  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
116, 10bitrd 245 . . . . . . . 8  |-  ( A  e.  N.  ->  ( A  <N  1o  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
1211biimpa 471 . . . . . . 7  |-  ( ( A  e.  N.  /\  A  <N  1o )  -> 
( A  e.  (/)  \/  A  =  (/) ) )
1312ord 367 . . . . . 6  |-  ( ( A  e.  N.  /\  A  <N  1o )  -> 
( -.  A  e.  (/)  ->  A  =  (/) ) )
143, 13mpi 17 . . . . 5  |-  ( ( A  e.  N.  /\  A  <N  1o )  ->  A  =  (/) )
1514ex 424 . . . 4  |-  ( A  e.  N.  ->  ( A  <N  1o  ->  A  =  (/) ) )
1615necon3ad 2634 . . 3  |-  ( A  e.  N.  ->  ( A  =/=  (/)  ->  -.  A  <N  1o ) )
172, 16mpd 15 . 2  |-  ( A  e.  N.  ->  -.  A  <N  1o )
18 ltrelpi 8755 . . . . 5  |-  <N  C_  ( N.  X.  N. )
1918brel 4917 . . . 4  |-  ( A 
<N  1o  ->  ( A  e.  N.  /\  1o  e.  N. ) )
2019simpld 446 . . 3  |-  ( A 
<N  1o  ->  A  e.  N. )
2120con3i 129 . 2  |-  ( -.  A  e.  N.  ->  -.  A  <N  1o )
2217, 21pm2.61i 158 1  |-  -.  A  <N  1o
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   (/)c0 3620   class class class wbr 4204   suc csuc 4575   omcom 4836   1oc1o 6708   N.cnpi 8708    <N clti 8711
This theorem is referenced by:  indpi  8773  pinq  8793  archnq  8846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-1o 6715  df-ni 8738  df-lti 8741
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