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Theorem nlt1pi 8788
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 8758 . . . 4  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
21simprbi 452 . . 3  |-  ( A  e.  N.  ->  A  =/=  (/) )
3 noel 3634 . . . . . 6  |-  -.  A  e.  (/)
4 1pi 8765 . . . . . . . . . 10  |-  1o  e.  N.
5 ltpiord 8769 . . . . . . . . . 10  |-  ( ( A  e.  N.  /\  1o  e.  N. )  -> 
( A  <N  1o  <->  A  e.  1o ) )
64, 5mpan2 654 . . . . . . . . 9  |-  ( A  e.  N.  ->  ( A  <N  1o  <->  A  e.  1o ) )
7 df-1o 6727 . . . . . . . . . . 11  |-  1o  =  suc  (/)
87eleq2i 2502 . . . . . . . . . 10  |-  ( A  e.  1o  <->  A  e.  suc  (/) )
9 elsucg 4651 . . . . . . . . . 10  |-  ( A  e.  N.  ->  ( A  e.  suc  (/)  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
108, 9syl5bb 250 . . . . . . . . 9  |-  ( A  e.  N.  ->  ( A  e.  1o  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
116, 10bitrd 246 . . . . . . . 8  |-  ( A  e.  N.  ->  ( A  <N  1o  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
1211biimpa 472 . . . . . . 7  |-  ( ( A  e.  N.  /\  A  <N  1o )  -> 
( A  e.  (/)  \/  A  =  (/) ) )
1312ord 368 . . . . . 6  |-  ( ( A  e.  N.  /\  A  <N  1o )  -> 
( -.  A  e.  (/)  ->  A  =  (/) ) )
143, 13mpi 17 . . . . 5  |-  ( ( A  e.  N.  /\  A  <N  1o )  ->  A  =  (/) )
1514ex 425 . . . 4  |-  ( A  e.  N.  ->  ( A  <N  1o  ->  A  =  (/) ) )
1615necon3ad 2639 . . 3  |-  ( A  e.  N.  ->  ( A  =/=  (/)  ->  -.  A  <N  1o ) )
172, 16mpd 15 . 2  |-  ( A  e.  N.  ->  -.  A  <N  1o )
18 ltrelpi 8771 . . . . 5  |-  <N  C_  ( N.  X.  N. )
1918brel 4929 . . . 4  |-  ( A 
<N  1o  ->  ( A  e.  N.  /\  1o  e.  N. ) )
2019simpld 447 . . 3  |-  ( A 
<N  1o  ->  A  e.  N. )
2120con3i 130 . 2  |-  ( -.  A  e.  N.  ->  -.  A  <N  1o )
2217, 21pm2.61i 159 1  |-  -.  A  <N  1o
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   (/)c0 3630   class class class wbr 4215   suc csuc 4586   omcom 4848   1oc1o 6720   N.cnpi 8724    <N clti 8727
This theorem is referenced by:  indpi  8789  pinq  8809  archnq  8862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-1o 6727  df-ni 8754  df-lti 8757
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