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Theorem nlt1pi 9071
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 9041 . . . 4  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
21simprbi 461 . . 3  |-  ( A  e.  N.  ->  A  =/=  (/) )
3 noel 3638 . . . . . 6  |-  -.  A  e.  (/)
4 1pi 9048 . . . . . . . . . 10  |-  1o  e.  N.
5 ltpiord 9052 . . . . . . . . . 10  |-  ( ( A  e.  N.  /\  1o  e.  N. )  -> 
( A  <N  1o  <->  A  e.  1o ) )
64, 5mpan2 666 . . . . . . . . 9  |-  ( A  e.  N.  ->  ( A  <N  1o  <->  A  e.  1o ) )
7 df-1o 6916 . . . . . . . . . . 11  |-  1o  =  suc  (/)
87eleq2i 2505 . . . . . . . . . 10  |-  ( A  e.  1o  <->  A  e.  suc  (/) )
9 elsucg 4782 . . . . . . . . . 10  |-  ( A  e.  N.  ->  ( A  e.  suc  (/)  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
108, 9syl5bb 257 . . . . . . . . 9  |-  ( A  e.  N.  ->  ( A  e.  1o  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
116, 10bitrd 253 . . . . . . . 8  |-  ( A  e.  N.  ->  ( A  <N  1o  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
1211biimpa 481 . . . . . . 7  |-  ( ( A  e.  N.  /\  A  <N  1o )  -> 
( A  e.  (/)  \/  A  =  (/) ) )
1312ord 377 . . . . . 6  |-  ( ( A  e.  N.  /\  A  <N  1o )  -> 
( -.  A  e.  (/)  ->  A  =  (/) ) )
143, 13mpi 17 . . . . 5  |-  ( ( A  e.  N.  /\  A  <N  1o )  ->  A  =  (/) )
1514ex 434 . . . 4  |-  ( A  e.  N.  ->  ( A  <N  1o  ->  A  =  (/) ) )
1615necon3ad 2642 . . 3  |-  ( A  e.  N.  ->  ( A  =/=  (/)  ->  -.  A  <N  1o ) )
172, 16mpd 15 . 2  |-  ( A  e.  N.  ->  -.  A  <N  1o )
18 ltrelpi 9054 . . . . 5  |-  <N  C_  ( N.  X.  N. )
1918brel 4883 . . . 4  |-  ( A 
<N  1o  ->  ( A  e.  N.  /\  1o  e.  N. ) )
2019simpld 456 . . 3  |-  ( A 
<N  1o  ->  A  e.  N. )
2120con3i 135 . 2  |-  ( -.  A  e.  N.  ->  -.  A  <N  1o )
2217, 21pm2.61i 164 1  |-  -.  A  <N  1o
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   (/)c0 3634   class class class wbr 4289   suc csuc 4717   omcom 6475   1oc1o 6909   N.cnpi 9007    <N clti 9010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-tr 4383  df-eprel 4628  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-om 6476  df-1o 6916  df-ni 9037  df-lti 9040
This theorem is referenced by:  indpi  9072  pinq  9092  archnq  9145
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