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Theorem elni2 7086
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 7081 . . 3  |-  ( A  e.  N.  ->  A  e.  om )
2 0npi 7085 . . . . . 6  |-  -.  (/)  e.  N.
3 eleq1 2178 . . . . . 6  |-  ( A  =  (/)  ->  ( A  e.  N.  <->  (/)  e.  N. ) )
42, 3mtbiri 647 . . . . 5  |-  ( A  =  (/)  ->  -.  A  e.  N. )
54con2i 599 . . . 4  |-  ( A  e.  N.  ->  -.  A  =  (/) )
6 0elnn 4500 . . . . . 6  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )
71, 6syl 14 . . . . 5  |-  ( A  e.  N.  ->  ( A  =  (/)  \/  (/)  e.  A
) )
87ord 696 . . . 4  |-  ( A  e.  N.  ->  ( -.  A  =  (/)  ->  (/)  e.  A
) )
95, 8mpd 13 . . 3  |-  ( A  e.  N.  ->  (/)  e.  A
)
101, 9jca 302 . 2  |-  ( A  e.  N.  ->  ( A  e.  om  /\  (/)  e.  A
) )
11 nndceq0 4499 . . . . . 6  |-  ( A  e.  om  -> DECID  A  =  (/) )
12 df-dc 803 . . . . . 6  |-  (DECID  A  =  (/) 
<->  ( A  =  (/)  \/ 
-.  A  =  (/) ) )
1311, 12sylib 121 . . . . 5  |-  ( A  e.  om  ->  ( A  =  (/)  \/  -.  A  =  (/) ) )
1413anim1i 336 . . . 4  |-  ( ( A  e.  om  /\  (/) 
e.  A )  -> 
( ( A  =  (/)  \/  -.  A  =  (/) )  /\  (/)  e.  A
) )
15 ancom 264 . . . . 5  |-  ( (
(/)  e.  A  /\  ( A  =  (/)  \/  -.  A  =  (/) ) )  <-> 
( ( A  =  (/)  \/  -.  A  =  (/) )  /\  (/)  e.  A
) )
16 andi 790 . . . . 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 3335 . . . . . . . . 9  |-  -.  (/)  e.  (/)
20 eleq2 2179 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( (/)  e.  A  <->  (/)  e.  (/) ) )
2119, 20mtbiri 647 . . . . . . . 8  |-  ( A  =  (/)  ->  -.  (/)  e.  A
)
2221pm2.21d 591 . . . . . . 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 2284 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
26 elni 7080 . . . . . . . 8  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
2726simplbi2 380 . . . . . . 7  |-  ( A  e.  om  ->  ( A  =/=  (/)  ->  A  e.  N. ) )
2825, 27syl5bir 152 . . . . . 6  |-  ( A  e.  om  ->  ( -.  A  =  (/)  ->  A  e.  N. ) )
2928adantld 274 . . . . 5  |-  ( A  e.  om  ->  (
( (/)  e.  A  /\  -.  A  =  (/) )  ->  A  e.  N. )
)
3024, 29jaod 689 . . . 4  |-  ( A  e.  om  ->  (
( ( (/)  e.  A  /\  A  =  (/) )  \/  ( (/)  e.  A  /\  -.  A  =  (/) ) )  ->  A  e.  N. ) )
3130adantr 272 . . 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 680  DECID wdc 802    = wceq 1314    e. wcel 1463    =/= wne 2283   (/)c0 3331   omcom 4472   N.cnpi 7044
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-suc 4261  df-iom 4473  df-ni 7076
This theorem is referenced by:  addclpi  7099  mulclpi  7100  mulcanpig  7107  addnidpig  7108  ltexpi  7109  ltmpig  7111  nnppipi  7115  archnqq  7189  enq0tr  7206
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