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Theorem elni2 7312
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 7307 . . 3  |-  ( A  e.  N.  ->  A  e.  om )
2 0npi 7311 . . . . . 6  |-  -.  (/)  e.  N.
3 eleq1 2240 . . . . . 6  |-  ( A  =  (/)  ->  ( A  e.  N.  <->  (/)  e.  N. ) )
42, 3mtbiri 675 . . . . 5  |-  ( A  =  (/)  ->  -.  A  e.  N. )
54con2i 627 . . . 4  |-  ( A  e.  N.  ->  -.  A  =  (/) )
6 0elnn 4618 . . . . . 6  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )
71, 6syl 14 . . . . 5  |-  ( A  e.  N.  ->  ( A  =  (/)  \/  (/)  e.  A
) )
87ord 724 . . . 4  |-  ( A  e.  N.  ->  ( -.  A  =  (/)  ->  (/)  e.  A
) )
95, 8mpd 13 . . 3  |-  ( A  e.  N.  ->  (/)  e.  A
)
101, 9jca 306 . 2  |-  ( A  e.  N.  ->  ( A  e.  om  /\  (/)  e.  A
) )
11 nndceq0 4617 . . . . . 6  |-  ( A  e.  om  -> DECID  A  =  (/) )
12 df-dc 835 . . . . . 6  |-  (DECID  A  =  (/) 
<->  ( A  =  (/)  \/ 
-.  A  =  (/) ) )
1311, 12sylib 122 . . . . 5  |-  ( A  e.  om  ->  ( A  =  (/)  \/  -.  A  =  (/) ) )
1413anim1i 340 . . . 4  |-  ( ( A  e.  om  /\  (/) 
e.  A )  -> 
( ( A  =  (/)  \/  -.  A  =  (/) )  /\  (/)  e.  A
) )
15 ancom 266 . . . . 5  |-  ( (
(/)  e.  A  /\  ( A  =  (/)  \/  -.  A  =  (/) ) )  <-> 
( ( A  =  (/)  \/  -.  A  =  (/) )  /\  (/)  e.  A
) )
16 andi 818 . . . . 5  |-  ( (
(/)  e.  A  /\  ( A  =  (/)  \/  -.  A  =  (/) ) )  <-> 
( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/)  e.  A  /\  -.  A  =  (/) ) ) )
1715, 16bitr3i 186 . . . 4  |-  ( ( ( A  =  (/)  \/ 
-.  A  =  (/) )  /\  (/)  e.  A )  <-> 
( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/)  e.  A  /\  -.  A  =  (/) ) ) )
1814, 17sylib 122 . . 3  |-  ( ( A  e.  om  /\  (/) 
e.  A )  -> 
( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/)  e.  A  /\  -.  A  =  (/) ) ) )
19 noel 3426 . . . . . . . . 9  |-  -.  (/)  e.  (/)
20 eleq2 2241 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( (/)  e.  A  <->  (/)  e.  (/) ) )
2119, 20mtbiri 675 . . . . . . . 8  |-  ( A  =  (/)  ->  -.  (/)  e.  A
)
2221pm2.21d 619 . . . . . . 7  |-  ( A  =  (/)  ->  ( (/)  e.  A  ->  A  e. 
N. ) )
2322impcom 125 . . . . . 6  |-  ( (
(/)  e.  A  /\  A  =  (/) )  ->  A  e.  N. )
2423a1i 9 . . . . 5  |-  ( A  e.  om  ->  (
( (/)  e.  A  /\  A  =  (/) )  ->  A  e.  N. )
)
25 df-ne 2348 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
26 elni 7306 . . . . . . . 8  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
2726simplbi2 385 . . . . . . 7  |-  ( A  e.  om  ->  ( A  =/=  (/)  ->  A  e.  N. ) )
2825, 27biimtrrid 153 . . . . . 6  |-  ( A  e.  om  ->  ( -.  A  =  (/)  ->  A  e.  N. ) )
2928adantld 278 . . . . 5  |-  ( A  e.  om  ->  (
( (/)  e.  A  /\  -.  A  =  (/) )  ->  A  e.  N. )
)
3024, 29jaod 717 . . . 4  |-  ( A  e.  om  ->  (
( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/)  e.  A  /\  -.  A  =  (/) ) )  ->  A  e.  N. ) )
3130adantr 276 . . 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 126 1  |-  ( A  e.  N.  <->  ( A  e.  om  /\  (/)  e.  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708  DECID wdc 834    = wceq 1353    e. wcel 2148    =/= wne 2347   (/)c0 3422   omcom 4589   N.cnpi 7270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-suc 4371  df-iom 4590  df-ni 7302
This theorem is referenced by:  addclpi  7325  mulclpi  7326  mulcanpig  7333  addnidpig  7334  ltexpi  7335  ltmpig  7337  nnppipi  7341  archnqq  7415  enq0tr  7432
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