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Theorem dfom2 4658
Description: An alternate definition of the set of natural numbers  om. Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol KI for the inner class builder of non-limit ordinal numbers (see nlimon 4642). (Contributed by NM, 1-Nov-2004.)
Assertion
Ref Expression
dfom2  |-  om  =  { x  e.  On  |  suc  x  C_  { y  e.  On  |  -.  Lim  y } }

Proof of Theorem dfom2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-om 4657 . 2  |-  om  =  { x  e.  On  |  A. z ( Lim  z  ->  x  e.  z ) }
2 onsssuc 4480 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  x  e.  On )  ->  ( z  C_  x  <->  z  e.  suc  x ) )
3 ontri1 4426 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  x  e.  On )  ->  ( z  C_  x  <->  -.  x  e.  z ) )
42, 3bitr3d 246 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  x  e.  On )  ->  ( z  e.  suc  x 
<->  -.  x  e.  z ) )
54ancoms 439 . . . . . . . . 9  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( z  e.  suc  x 
<->  -.  x  e.  z ) )
6 limeq 4404 . . . . . . . . . . . 12  |-  ( y  =  z  ->  ( Lim  y  <->  Lim  z ) )
76notbid 285 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( -.  Lim  y  <->  -.  Lim  z
) )
87elrab 2923 . . . . . . . . . 10  |-  ( z  e.  { y  e.  On  |  -.  Lim  y }  <->  ( z  e.  On  /\  -.  Lim  z ) )
98a1i 10 . . . . . . . . 9  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( z  e.  {
y  e.  On  |  -.  Lim  y }  <->  ( z  e.  On  /\  -.  Lim  z ) ) )
105, 9imbi12d 311 . . . . . . . 8  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( ( z  e. 
suc  x  ->  z  e.  { y  e.  On  |  -.  Lim  y } )  <->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
1110pm5.74da 668 . . . . . . 7  |-  ( x  e.  On  ->  (
( z  e.  On  ->  ( z  e.  suc  x  ->  z  e.  {
y  e.  On  |  -.  Lim  y } ) )  <->  ( z  e.  On  ->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) ) )
12 vex 2791 . . . . . . . . . . 11  |-  z  e. 
_V
13 limelon 4455 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  z  e.  On )
1412, 13mpan 651 . . . . . . . . . 10  |-  ( Lim  z  ->  z  e.  On )
1514pm4.71ri 614 . . . . . . . . 9  |-  ( Lim  z  <->  ( z  e.  On  /\  Lim  z
) )
1615imbi1i 315 . . . . . . . 8  |-  ( ( Lim  z  ->  x  e.  z )  <->  ( (
z  e.  On  /\  Lim  z )  ->  x  e.  z ) )
17 impexp 433 . . . . . . . 8  |-  ( ( ( z  e.  On  /\ 
Lim  z )  ->  x  e.  z )  <->  ( z  e.  On  ->  ( Lim  z  ->  x  e.  z ) ) )
18 con34b 283 . . . . . . . . . 10  |-  ( ( Lim  z  ->  x  e.  z )  <->  ( -.  x  e.  z  ->  -. 
Lim  z ) )
19 ibar 490 . . . . . . . . . . 11  |-  ( z  e.  On  ->  ( -.  Lim  z  <->  ( z  e.  On  /\  -.  Lim  z ) ) )
2019imbi2d 307 . . . . . . . . . 10  |-  ( z  e.  On  ->  (
( -.  x  e.  z  ->  -.  Lim  z
)  <->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2118, 20syl5bb 248 . . . . . . . . 9  |-  ( z  e.  On  ->  (
( Lim  z  ->  x  e.  z )  <->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2221pm5.74i 236 . . . . . . . 8  |-  ( ( z  e.  On  ->  ( Lim  z  ->  x  e.  z ) )  <->  ( z  e.  On  ->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2316, 17, 223bitri 262 . . . . . . 7  |-  ( ( Lim  z  ->  x  e.  z )  <->  ( z  e.  On  ->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2411, 23syl6rbbr 255 . . . . . 6  |-  ( x  e.  On  ->  (
( Lim  z  ->  x  e.  z )  <->  ( z  e.  On  ->  ( z  e.  suc  x  ->  z  e.  { y  e.  On  |  -.  Lim  y } ) ) ) )
25 impexp 433 . . . . . . 7  |-  ( ( ( z  e.  On  /\  z  e.  suc  x
)  ->  z  e.  { y  e.  On  |  -.  Lim  y } )  <-> 
( z  e.  On  ->  ( z  e.  suc  x  ->  z  e.  {
y  e.  On  |  -.  Lim  y } ) ) )
26 simpr 447 . . . . . . . . 9  |-  ( ( z  e.  On  /\  z  e.  suc  x )  ->  z  e.  suc  x )
27 suceloni 4604 . . . . . . . . . . 11  |-  ( x  e.  On  ->  suc  x  e.  On )
28 onelon 4417 . . . . . . . . . . . 12  |-  ( ( suc  x  e.  On  /\  z  e.  suc  x
)  ->  z  e.  On )
2928ex 423 . . . . . . . . . . 11  |-  ( suc  x  e.  On  ->  ( z  e.  suc  x  ->  z  e.  On ) )
3027, 29syl 15 . . . . . . . . . 10  |-  ( x  e.  On  ->  (
z  e.  suc  x  ->  z  e.  On ) )
3130ancrd 537 . . . . . . . . 9  |-  ( x  e.  On  ->  (
z  e.  suc  x  ->  ( z  e.  On  /\  z  e.  suc  x
) ) )
3226, 31impbid2 195 . . . . . . . 8  |-  ( x  e.  On  ->  (
( z  e.  On  /\  z  e.  suc  x
)  <->  z  e.  suc  x ) )
3332imbi1d 308 . . . . . . 7  |-  ( x  e.  On  ->  (
( ( z  e.  On  /\  z  e. 
suc  x )  -> 
z  e.  { y  e.  On  |  -.  Lim  y } )  <->  ( z  e.  suc  x  ->  z  e.  { y  e.  On  |  -.  Lim  y } ) ) )
3425, 33syl5bbr 250 . . . . . 6  |-  ( x  e.  On  ->  (
( z  e.  On  ->  ( z  e.  suc  x  ->  z  e.  {
y  e.  On  |  -.  Lim  y } ) )  <->  ( z  e. 
suc  x  ->  z  e.  { y  e.  On  |  -.  Lim  y } ) ) )
3524, 34bitrd 244 . . . . 5  |-  ( x  e.  On  ->  (
( Lim  z  ->  x  e.  z )  <->  ( z  e.  suc  x  ->  z  e.  { y  e.  On  |  -.  Lim  y } ) ) )
3635albidv 1611 . . . 4  |-  ( x  e.  On  ->  ( A. z ( Lim  z  ->  x  e.  z )  <->  A. z ( z  e. 
suc  x  ->  z  e.  { y  e.  On  |  -.  Lim  y } ) ) )
37 dfss2 3169 . . . 4  |-  ( suc  x  C_  { y  e.  On  |  -.  Lim  y }  <->  A. z ( z  e.  suc  x  -> 
z  e.  { y  e.  On  |  -.  Lim  y } ) )
3836, 37syl6bbr 254 . . 3  |-  ( x  e.  On  ->  ( A. z ( Lim  z  ->  x  e.  z )  <->  suc  x  C_  { y  e.  On  |  -.  Lim  y } ) )
3938rabbiia 2778 . 2  |-  { x  e.  On  |  A. z
( Lim  z  ->  x  e.  z ) }  =  { x  e.  On  |  suc  x  C_ 
{ y  e.  On  |  -.  Lim  y } }
401, 39eqtri 2303 1  |-  om  =  { x  e.  On  |  suc  x  C_  { y  e.  On  |  -.  Lim  y } }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   Oncon0 4392   Lim wlim 4393   suc csuc 4394   omcom 4656
This theorem is referenced by:  omsson  4660
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657
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