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Theorem elni2 6566
Description: Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
Assertion
Ref Expression
elni2  |-  ( A  e.  N.  <->  ( A  e.  om  /\  (/)  e.  A
) )

Proof of Theorem elni2
StepHypRef Expression
1 pinn 6561 . . 3  |-  ( A  e.  N.  ->  A  e.  om )
2 0npi 6565 . . . . . 6  |-  -.  (/)  e.  N.
3 eleq1 2142 . . . . . 6  |-  ( A  =  (/)  ->  ( A  e.  N.  <->  (/)  e.  N. ) )
42, 3mtbiri 633 . . . . 5  |-  ( A  =  (/)  ->  -.  A  e.  N. )
54con2i 590 . . . 4  |-  ( A  e.  N.  ->  -.  A  =  (/) )
6 0elnn 4366 . . . . . 6  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )
71, 6syl 14 . . . . 5  |-  ( A  e.  N.  ->  ( A  =  (/)  \/  (/)  e.  A
) )
87ord 676 . . . 4  |-  ( A  e.  N.  ->  ( -.  A  =  (/)  ->  (/)  e.  A
) )
95, 8mpd 13 . . 3  |-  ( A  e.  N.  ->  (/)  e.  A
)
101, 9jca 300 . 2  |-  ( A  e.  N.  ->  ( A  e.  om  /\  (/)  e.  A
) )
11 nndceq0 4365 . . . . . 6  |-  ( A  e.  om  -> DECID  A  =  (/) )
12 df-dc 777 . . . . . 6  |-  (DECID  A  =  (/) 
<->  ( A  =  (/)  \/ 
-.  A  =  (/) ) )
1311, 12sylib 120 . . . . 5  |-  ( A  e.  om  ->  ( A  =  (/)  \/  -.  A  =  (/) ) )
1413anim1i 333 . . . 4  |-  ( ( A  e.  om  /\  (/) 
e.  A )  -> 
( ( A  =  (/)  \/  -.  A  =  (/) )  /\  (/)  e.  A
) )
15 ancom 262 . . . . 5  |-  ( (
(/)  e.  A  /\  ( A  =  (/)  \/  -.  A  =  (/) ) )  <-> 
( ( A  =  (/)  \/  -.  A  =  (/) )  /\  (/)  e.  A
) )
16 andi 765 . . . . 5  |-  ( (
(/)  e.  A  /\  ( A  =  (/)  \/  -.  A  =  (/) ) )  <-> 
( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/)  e.  A  /\  -.  A  =  (/) ) ) )
1715, 16bitr3i 184 . . . 4  |-  ( ( ( A  =  (/)  \/ 
-.  A  =  (/) )  /\  (/)  e.  A )  <-> 
( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/)  e.  A  /\  -.  A  =  (/) ) ) )
1814, 17sylib 120 . . 3  |-  ( ( A  e.  om  /\  (/) 
e.  A )  -> 
( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/)  e.  A  /\  -.  A  =  (/) ) ) )
19 noel 3262 . . . . . . . . 9  |-  -.  (/)  e.  (/)
20 eleq2 2143 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( (/)  e.  A  <->  (/)  e.  (/) ) )
2119, 20mtbiri 633 . . . . . . . 8  |-  ( A  =  (/)  ->  -.  (/)  e.  A
)
2221pm2.21d 582 . . . . . . 7  |-  ( A  =  (/)  ->  ( (/)  e.  A  ->  A  e. 
N. ) )
2322impcom 123 . . . . . 6  |-  ( (
(/)  e.  A  /\  A  =  (/) )  ->  A  e.  N. )
2423a1i 9 . . . . 5  |-  ( A  e.  om  ->  (
( (/)  e.  A  /\  A  =  (/) )  ->  A  e.  N. )
)
25 df-ne 2247 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
26 elni 6560 . . . . . . . 8  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
2726simplbi2 377 . . . . . . 7  |-  ( A  e.  om  ->  ( A  =/=  (/)  ->  A  e.  N. ) )
2825, 27syl5bir 151 . . . . . 6  |-  ( A  e.  om  ->  ( -.  A  =  (/)  ->  A  e.  N. ) )
2928adantld 272 . . . . 5  |-  ( A  e.  om  ->  (
( (/)  e.  A  /\  -.  A  =  (/) )  ->  A  e.  N. )
)
3024, 29jaod 670 . . . 4  |-  ( A  e.  om  ->  (
( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/)  e.  A  /\  -.  A  =  (/) ) )  ->  A  e.  N. ) )
3130adantr 270 . . 3  |-  ( ( A  e.  om  /\  (/) 
e.  A )  -> 
( ( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/) 
e.  A  /\  -.  A  =  (/) ) )  ->  A  e.  N. ) )
3218, 31mpd 13 . 2  |-  ( ( A  e.  om  /\  (/) 
e.  A )  ->  A  e.  N. )
3310, 32impbii 124 1  |-  ( A  e.  N.  <->  ( A  e.  om  /\  (/)  e.  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662  DECID wdc 776    = wceq 1285    e. wcel 1434    =/= wne 2246   (/)c0 3258   omcom 4339   N.cnpi 6524
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  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  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-uni 3610  df-int 3645  df-suc 4134  df-iom 4340  df-ni 6556
This theorem is referenced by:  addclpi  6579  mulclpi  6580  mulcanpig  6587  addnidpig  6588  ltexpi  6589  ltmpig  6591  nnppipi  6595  archnqq  6669  enq0tr  6686
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