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Theorem elni2 7090
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 7085 . . 3  |-  ( A  e.  N.  ->  A  e.  om )
2 0npi 7089 . . . . . 6  |-  -.  (/)  e.  N.
3 eleq1 2180 . . . . . 6  |-  ( A  =  (/)  ->  ( A  e.  N.  <->  (/)  e.  N. ) )
42, 3mtbiri 649 . . . . 5  |-  ( A  =  (/)  ->  -.  A  e.  N. )
54con2i 601 . . . 4  |-  ( A  e.  N.  ->  -.  A  =  (/) )
6 0elnn 4502 . . . . . 6  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )
71, 6syl 14 . . . . 5  |-  ( A  e.  N.  ->  ( A  =  (/)  \/  (/)  e.  A
) )
87ord 698 . . . 4  |-  ( A  e.  N.  ->  ( -.  A  =  (/)  ->  (/)  e.  A
) )
95, 8mpd 13 . . 3  |-  ( A  e.  N.  ->  (/)  e.  A
)
101, 9jca 304 . 2  |-  ( A  e.  N.  ->  ( A  e.  om  /\  (/)  e.  A
) )
11 nndceq0 4501 . . . . . 6  |-  ( A  e.  om  -> DECID  A  =  (/) )
12 df-dc 805 . . . . . 6  |-  (DECID  A  =  (/) 
<->  ( A  =  (/)  \/ 
-.  A  =  (/) ) )
1311, 12sylib 121 . . . . 5  |-  ( A  e.  om  ->  ( A  =  (/)  \/  -.  A  =  (/) ) )
1413anim1i 338 . . . 4  |-  ( ( A  e.  om  /\  (/) 
e.  A )  -> 
( ( A  =  (/)  \/  -.  A  =  (/) )  /\  (/)  e.  A
) )
15 ancom 264 . . . . 5  |-  ( (
(/)  e.  A  /\  ( A  =  (/)  \/  -.  A  =  (/) ) )  <-> 
( ( A  =  (/)  \/  -.  A  =  (/) )  /\  (/)  e.  A
) )
16 andi 792 . . . . 5  |-  ( (
(/)  e.  A  /\  ( A  =  (/)  \/  -.  A  =  (/) ) )  <-> 
( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/)  e.  A  /\  -.  A  =  (/) ) ) )
1715, 16bitr3i 185 . . . 4  |-  ( ( ( A  =  (/)  \/ 
-.  A  =  (/) )  /\  (/)  e.  A )  <-> 
( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/)  e.  A  /\  -.  A  =  (/) ) ) )
1814, 17sylib 121 . . 3  |-  ( ( A  e.  om  /\  (/) 
e.  A )  -> 
( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/)  e.  A  /\  -.  A  =  (/) ) ) )
19 noel 3337 . . . . . . . . 9  |-  -.  (/)  e.  (/)
20 eleq2 2181 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( (/)  e.  A  <->  (/)  e.  (/) ) )
2119, 20mtbiri 649 . . . . . . . 8  |-  ( A  =  (/)  ->  -.  (/)  e.  A
)
2221pm2.21d 593 . . . . . . 7  |-  ( A  =  (/)  ->  ( (/)  e.  A  ->  A  e. 
N. ) )
2322impcom 124 . . . . . 6  |-  ( (
(/)  e.  A  /\  A  =  (/) )  ->  A  e.  N. )
2423a1i 9 . . . . 5  |-  ( A  e.  om  ->  (
( (/)  e.  A  /\  A  =  (/) )  ->  A  e.  N. )
)
25 df-ne 2286 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
26 elni 7084 . . . . . . . 8  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
2726simplbi2 382 . . . . . . 7  |-  ( A  e.  om  ->  ( A  =/=  (/)  ->  A  e.  N. ) )
2825, 27syl5bir 152 . . . . . 6  |-  ( A  e.  om  ->  ( -.  A  =  (/)  ->  A  e.  N. ) )
2928adantld 276 . . . . 5  |-  ( A  e.  om  ->  (
( (/)  e.  A  /\  -.  A  =  (/) )  ->  A  e.  N. )
)
3024, 29jaod 691 . . . 4  |-  ( A  e.  om  ->  (
( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/)  e.  A  /\  -.  A  =  (/) ) )  ->  A  e.  N. ) )
3130adantr 274 . . 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 125 1  |-  ( A  e.  N.  <->  ( A  e.  om  /\  (/)  e.  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 682  DECID wdc 804    = wceq 1316    e. wcel 1465    =/= wne 2285   (/)c0 3333   omcom 4474   N.cnpi 7048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-suc 4263  df-iom 4475  df-ni 7080
This theorem is referenced by:  addclpi  7103  mulclpi  7104  mulcanpig  7111  addnidpig  7112  ltexpi  7113  ltmpig  7115  nnppipi  7119  archnqq  7193  enq0tr  7210
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