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Theorem nlt1pig 6593
Description: No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
Assertion
Ref Expression
nlt1pig  |-  ( A  e.  N.  ->  -.  A  <N  1o )

Proof of Theorem nlt1pig
StepHypRef Expression
1 elni 6560 . . 3  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
21simprbi 269 . 2  |-  ( A  e.  N.  ->  A  =/=  (/) )
3 noel 3262 . . . . 5  |-  -.  A  e.  (/)
4 1pi 6567 . . . . . . . . 9  |-  1o  e.  N.
5 ltpiord 6571 . . . . . . . . 9  |-  ( ( A  e.  N.  /\  1o  e.  N. )  -> 
( A  <N  1o  <->  A  e.  1o ) )
64, 5mpan2 416 . . . . . . . 8  |-  ( A  e.  N.  ->  ( A  <N  1o  <->  A  e.  1o ) )
7 df-1o 6065 . . . . . . . . . 10  |-  1o  =  suc  (/)
87eleq2i 2146 . . . . . . . . 9  |-  ( A  e.  1o  <->  A  e.  suc  (/) )
9 elsucg 4167 . . . . . . . . 9  |-  ( A  e.  N.  ->  ( A  e.  suc  (/)  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
108, 9syl5bb 190 . . . . . . . 8  |-  ( A  e.  N.  ->  ( A  e.  1o  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
116, 10bitrd 186 . . . . . . 7  |-  ( A  e.  N.  ->  ( A  <N  1o  <->  ( A  e.  (/)  \/  A  =  (/) ) ) )
1211biimpa 290 . . . . . 6  |-  ( ( A  e.  N.  /\  A  <N  1o )  -> 
( A  e.  (/)  \/  A  =  (/) ) )
1312ord 676 . . . . 5  |-  ( ( A  e.  N.  /\  A  <N  1o )  -> 
( -.  A  e.  (/)  ->  A  =  (/) ) )
143, 13mpi 15 . . . 4  |-  ( ( A  e.  N.  /\  A  <N  1o )  ->  A  =  (/) )
1514ex 113 . . 3  |-  ( A  e.  N.  ->  ( A  <N  1o  ->  A  =  (/) ) )
1615necon3ad 2288 . 2  |-  ( A  e.  N.  ->  ( A  =/=  (/)  ->  -.  A  <N  1o ) )
172, 16mpd 13 1  |-  ( A  e.  N.  ->  -.  A  <N  1o )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434    =/= wne 2246   (/)c0 3258   class class class wbr 3793   suc csuc 4128   omcom 4339   1oc1o 6058   N.cnpi 6524    <N clti 6527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-eprel 4052  df-suc 4134  df-iom 4340  df-xp 4377  df-1o 6065  df-ni 6556  df-lti 6559
This theorem is referenced by:  caucvgsr  7040
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