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Theorem dfom2 4549
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 4533). (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
StepHypRef Expression
1 df-om 4548 . 2  |-  om  =  { x  e.  On  |  A. z ( Lim  z  ->  x  e.  z ) }
2 onsssuc 4373 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  x  e.  On )  ->  ( z  C_  x  <->  z  e.  suc  x ) )
3 ontri1 4319 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  x  e.  On )  ->  ( z  C_  x  <->  -.  x  e.  z ) )
42, 3bitr3d 248 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  x  e.  On )  ->  ( z  e.  suc  x 
<->  -.  x  e.  z ) )
54ancoms 441 . . . . . . . . 9  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( z  e.  suc  x 
<->  -.  x  e.  z ) )
6 limeq 4297 . . . . . . . . . . . 12  |-  ( y  =  z  ->  ( Lim  y  <->  Lim  z ) )
76notbid 287 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( -.  Lim  y  <->  -.  Lim  z
) )
87elrab 2860 . . . . . . . . . 10  |-  ( z  e.  { y  e.  On  |  -.  Lim  y }  <->  ( z  e.  On  /\  -.  Lim  z ) )
98a1i 12 . . . . . . . . 9  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( z  e.  {
y  e.  On  |  -.  Lim  y }  <->  ( z  e.  On  /\  -.  Lim  z ) ) )
105, 9imbi12d 313 . . . . . . . 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 671 . . . . . . 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 2730 . . . . . . . . . . 11  |-  z  e. 
_V
13 limelon 4348 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  z  e.  On )
1412, 13mpan 654 . . . . . . . . . 10  |-  ( Lim  z  ->  z  e.  On )
1514pm4.71ri 617 . . . . . . . . 9  |-  ( Lim  z  <->  ( z  e.  On  /\  Lim  z
) )
1615imbi1i 317 . . . . . . . 8  |-  ( ( Lim  z  ->  x  e.  z )  <->  ( (
z  e.  On  /\  Lim  z )  ->  x  e.  z ) )
17 impexp 435 . . . . . . . 8  |-  ( ( ( z  e.  On  /\ 
Lim  z )  ->  x  e.  z )  <->  ( z  e.  On  ->  ( Lim  z  ->  x  e.  z ) ) )
18 con34b 285 . . . . . . . . . 10  |-  ( ( Lim  z  ->  x  e.  z )  <->  ( -.  x  e.  z  ->  -. 
Lim  z ) )
19 ibar 492 . . . . . . . . . . 11  |-  ( z  e.  On  ->  ( -.  Lim  z  <->  ( z  e.  On  /\  -.  Lim  z ) ) )
2019imbi2d 309 . . . . . . . . . 10  |-  ( z  e.  On  ->  (
( -.  x  e.  z  ->  -.  Lim  z
)  <->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2118, 20syl5bb 250 . . . . . . . . 9  |-  ( z  e.  On  ->  (
( Lim  z  ->  x  e.  z )  <->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2221pm5.74i 238 . . . . . . . 8  |-  ( ( z  e.  On  ->  ( Lim  z  ->  x  e.  z ) )  <->  ( z  e.  On  ->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2316, 17, 223bitri 264 . . . . . . 7  |-  ( ( Lim  z  ->  x  e.  z )  <->  ( z  e.  On  ->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2411, 23syl6rbbr 257 . . . . . 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 435 . . . . . . 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 449 . . . . . . . . 9  |-  ( ( z  e.  On  /\  z  e.  suc  x )  ->  z  e.  suc  x )
27 suceloni 4495 . . . . . . . . . . 11  |-  ( x  e.  On  ->  suc  x  e.  On )
28 onelon 4310 . . . . . . . . . . . 12  |-  ( ( suc  x  e.  On  /\  z  e.  suc  x
)  ->  z  e.  On )
2928ex 425 . . . . . . . . . . 11  |-  ( suc  x  e.  On  ->  ( z  e.  suc  x  ->  z  e.  On ) )
3027, 29syl 17 . . . . . . . . . 10  |-  ( x  e.  On  ->  (
z  e.  suc  x  ->  z  e.  On ) )
3130ancrd 539 . . . . . . . . 9  |-  ( x  e.  On  ->  (
z  e.  suc  x  ->  ( z  e.  On  /\  z  e.  suc  x
) ) )
3226, 31impbid2 197 . . . . . . . 8  |-  ( x  e.  On  ->  (
( z  e.  On  /\  z  e.  suc  x
)  <->  z  e.  suc  x ) )
3332imbi1d 310 . . . . . . 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 252 . . . . . 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 246 . . . . 5  |-  ( x  e.  On  ->  (
( Lim  z  ->  x  e.  z )  <->  ( z  e.  suc  x  ->  z  e.  { y  e.  On  |  -.  Lim  y } ) ) )
3635albidv 2004 . . . 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 3092 . . . 4  |-  ( suc  x  C_  { y  e.  On  |  -.  Lim  y }  <->  A. z ( z  e.  suc  x  -> 
z  e.  { y  e.  On  |  -.  Lim  y } ) )
3836, 37syl6bbr 256 . . 3  |-  ( x  e.  On  ->  ( A. z ( Lim  z  ->  x  e.  z )  <->  suc  x  C_  { y  e.  On  |  -.  Lim  y } ) )
3938rabbiia 2717 . 2  |-  { x  e.  On  |  A. z
( Lim  z  ->  x  e.  z ) }  =  { x  e.  On  |  suc  x  C_ 
{ y  e.  On  |  -.  Lim  y } }
401, 39eqtri 2273 1  |-  om  =  { x  e.  On  |  suc  x  C_  { y  e.  On  |  -.  Lim  y } }
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532    = wceq 1619    e. wcel 1621   {crab 2512   _Vcvv 2727    C_ wss 3078   Oncon0 4285   Lim wlim 4286   suc csuc 4287   omcom 4547
This theorem is referenced by:  omsson  4551
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548
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