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Theorem nlt1pi 9331
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 9301 . . . 4  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
21simprbi 466 . . 3  |-  ( A  e.  N.  ->  A  =/=  (/) )
3 noel 3735 . . . . . 6  |-  -.  A  e.  (/)
4 1pi 9308 . . . . . . . . . 10  |-  1o  e.  N.
5 ltpiord 9312 . . . . . . . . . 10  |-  ( ( A  e.  N.  /\  1o  e.  N. )  -> 
( A  <N  1o  <->  A  e.  1o ) )
64, 5mpan2 677 . . . . . . . . 9  |-  ( A  e.  N.  ->  ( A  <N  1o  <->  A  e.  1o ) )
7 df-1o 7182 . . . . . . . . . . 11  |-  1o  =  suc  (/)
87eleq2i 2521 . . . . . . . . . 10  |-  ( A  e.  1o  <->  A  e.  suc  (/) )
9 elsucg 5490 . . . . . . . . . 10  |-  ( A  e.  N.  ->  ( A  e.  suc  (/)  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
108, 9syl5bb 261 . . . . . . . . 9  |-  ( A  e.  N.  ->  ( A  e.  1o  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
116, 10bitrd 257 . . . . . . . 8  |-  ( A  e.  N.  ->  ( A  <N  1o  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
1211biimpa 487 . . . . . . 7  |-  ( ( A  e.  N.  /\  A  <N  1o )  -> 
( A  e.  (/)  \/  A  =  (/) ) )
1312ord 379 . . . . . 6  |-  ( ( A  e.  N.  /\  A  <N  1o )  -> 
( -.  A  e.  (/)  ->  A  =  (/) ) )
143, 13mpi 20 . . . . 5  |-  ( ( A  e.  N.  /\  A  <N  1o )  ->  A  =  (/) )
1514ex 436 . . . 4  |-  ( A  e.  N.  ->  ( A  <N  1o  ->  A  =  (/) ) )
1615necon3ad 2637 . . 3  |-  ( A  e.  N.  ->  ( A  =/=  (/)  ->  -.  A  <N  1o ) )
172, 16mpd 15 . 2  |-  ( A  e.  N.  ->  -.  A  <N  1o )
18 ltrelpi 9314 . . . . 5  |-  <N  C_  ( N.  X.  N. )
1918brel 4883 . . . 4  |-  ( A 
<N  1o  ->  ( A  e.  N.  /\  1o  e.  N. ) )
2019simpld 461 . . 3  |-  ( A 
<N  1o  ->  A  e.  N. )
2120con3i 141 . 2  |-  ( -.  A  e.  N.  ->  -.  A  <N  1o )
2217, 21pm2.61i 168 1  |-  -.  A  <N  1o
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   (/)c0 3731   class class class wbr 4402   suc csuc 5425   omcom 6692   1oc1o 7175   N.cnpi 9269    <N clti 9272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-om 6693  df-1o 7182  df-ni 9297  df-lti 9300
This theorem is referenced by:  indpi  9332  pinq  9352  archnq  9405
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