MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfom2 Structured version   Unicode version

Theorem dfom2 6697
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 6681). (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 6696 . 2  |-  om  =  { x  e.  On  |  A. z ( Lim  z  ->  x  e.  z ) }
2 onsssuc 4971 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  x  e.  On )  ->  ( z  C_  x  <->  z  e.  suc  x ) )
3 ontri1 4918 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  x  e.  On )  ->  ( z  C_  x  <->  -.  x  e.  z ) )
42, 3bitr3d 255 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  x  e.  On )  ->  ( z  e.  suc  x 
<->  -.  x  e.  z ) )
54ancoms 453 . . . . . . . . 9  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( z  e.  suc  x 
<->  -.  x  e.  z ) )
6 limeq 4896 . . . . . . . . . . . 12  |-  ( y  =  z  ->  ( Lim  y  <->  Lim  z ) )
76notbid 294 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( -.  Lim  y  <->  -.  Lim  z
) )
87elrab 3266 . . . . . . . . . 10  |-  ( z  e.  { y  e.  On  |  -.  Lim  y }  <->  ( z  e.  On  /\  -.  Lim  z ) )
98a1i 11 . . . . . . . . 9  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( z  e.  {
y  e.  On  |  -.  Lim  y }  <->  ( z  e.  On  /\  -.  Lim  z ) ) )
105, 9imbi12d 320 . . . . . . . 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 687 . . . . . . 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 3121 . . . . . . . . . . 11  |-  z  e. 
_V
13 limelon 4947 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  z  e.  On )
1412, 13mpan 670 . . . . . . . . . 10  |-  ( Lim  z  ->  z  e.  On )
1514pm4.71ri 633 . . . . . . . . 9  |-  ( Lim  z  <->  ( z  e.  On  /\  Lim  z
) )
1615imbi1i 325 . . . . . . . 8  |-  ( ( Lim  z  ->  x  e.  z )  <->  ( (
z  e.  On  /\  Lim  z )  ->  x  e.  z ) )
17 impexp 446 . . . . . . . 8  |-  ( ( ( z  e.  On  /\ 
Lim  z )  ->  x  e.  z )  <->  ( z  e.  On  ->  ( Lim  z  ->  x  e.  z ) ) )
18 con34b 292 . . . . . . . . . 10  |-  ( ( Lim  z  ->  x  e.  z )  <->  ( -.  x  e.  z  ->  -. 
Lim  z ) )
19 ibar 504 . . . . . . . . . . 11  |-  ( z  e.  On  ->  ( -.  Lim  z  <->  ( z  e.  On  /\  -.  Lim  z ) ) )
2019imbi2d 316 . . . . . . . . . 10  |-  ( z  e.  On  ->  (
( -.  x  e.  z  ->  -.  Lim  z
)  <->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2118, 20syl5bb 257 . . . . . . . . 9  |-  ( z  e.  On  ->  (
( Lim  z  ->  x  e.  z )  <->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2221pm5.74i 245 . . . . . . . 8  |-  ( ( z  e.  On  ->  ( Lim  z  ->  x  e.  z ) )  <->  ( z  e.  On  ->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2316, 17, 223bitri 271 . . . . . . 7  |-  ( ( Lim  z  ->  x  e.  z )  <->  ( z  e.  On  ->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2411, 23syl6rbbr 264 . . . . . 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 446 . . . . . . 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 461 . . . . . . . . 9  |-  ( ( z  e.  On  /\  z  e.  suc  x )  ->  z  e.  suc  x )
27 suceloni 6643 . . . . . . . . . . 11  |-  ( x  e.  On  ->  suc  x  e.  On )
28 onelon 4909 . . . . . . . . . . . 12  |-  ( ( suc  x  e.  On  /\  z  e.  suc  x
)  ->  z  e.  On )
2928ex 434 . . . . . . . . . . 11  |-  ( suc  x  e.  On  ->  ( z  e.  suc  x  ->  z  e.  On ) )
3027, 29syl 16 . . . . . . . . . 10  |-  ( x  e.  On  ->  (
z  e.  suc  x  ->  z  e.  On ) )
3130ancrd 554 . . . . . . . . 9  |-  ( x  e.  On  ->  (
z  e.  suc  x  ->  ( z  e.  On  /\  z  e.  suc  x
) ) )
3226, 31impbid2 204 . . . . . . . 8  |-  ( x  e.  On  ->  (
( z  e.  On  /\  z  e.  suc  x
)  <->  z  e.  suc  x ) )
3332imbi1d 317 . . . . . . 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 259 . . . . . 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 253 . . . . 5  |-  ( x  e.  On  ->  (
( Lim  z  ->  x  e.  z )  <->  ( z  e.  suc  x  ->  z  e.  { y  e.  On  |  -.  Lim  y } ) ) )
3635albidv 1689 . . . 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 3498 . . . 4  |-  ( suc  x  C_  { y  e.  On  |  -.  Lim  y }  <->  A. z ( z  e.  suc  x  -> 
z  e.  { y  e.  On  |  -.  Lim  y } ) )
3836, 37syl6bbr 263 . . 3  |-  ( x  e.  On  ->  ( A. z ( Lim  z  ->  x  e.  z )  <->  suc  x  C_  { y  e.  On  |  -.  Lim  y } ) )
3938rabbiia 3107 . 2  |-  { x  e.  On  |  A. z
( Lim  z  ->  x  e.  z ) }  =  { x  e.  On  |  suc  x  C_ 
{ y  e.  On  |  -.  Lim  y } }
401, 39eqtri 2496 1  |-  om  =  { x  e.  On  |  suc  x  C_  { y  e.  On  |  -.  Lim  y } }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118    C_ wss 3481   Oncon0 4884   Lim wlim 4885   suc csuc 4886   omcom 6695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-om 6696
This theorem is referenced by:  omsson  6699
  Copyright terms: Public domain W3C validator