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

Theorem dfom2 4838
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 4822). (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 4837 . 2  |-  om  =  { x  e.  On  |  A. z ( Lim  z  ->  x  e.  z ) }
2 onsssuc 4660 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  x  e.  On )  ->  ( z  C_  x  <->  z  e.  suc  x ) )
3 ontri1 4607 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  x  e.  On )  ->  ( z  C_  x  <->  -.  x  e.  z ) )
42, 3bitr3d 247 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  x  e.  On )  ->  ( z  e.  suc  x 
<->  -.  x  e.  z ) )
54ancoms 440 . . . . . . . . 9  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( z  e.  suc  x 
<->  -.  x  e.  z ) )
6 limeq 4585 . . . . . . . . . . . 12  |-  ( y  =  z  ->  ( Lim  y  <->  Lim  z ) )
76notbid 286 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( -.  Lim  y  <->  -.  Lim  z
) )
87elrab 3084 . . . . . . . . . 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 312 . . . . . . . 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 669 . . . . . . 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 2951 . . . . . . . . . . 11  |-  z  e. 
_V
13 limelon 4636 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  z  e.  On )
1412, 13mpan 652 . . . . . . . . . 10  |-  ( Lim  z  ->  z  e.  On )
1514pm4.71ri 615 . . . . . . . . 9  |-  ( Lim  z  <->  ( z  e.  On  /\  Lim  z
) )
1615imbi1i 316 . . . . . . . 8  |-  ( ( Lim  z  ->  x  e.  z )  <->  ( (
z  e.  On  /\  Lim  z )  ->  x  e.  z ) )
17 impexp 434 . . . . . . . 8  |-  ( ( ( z  e.  On  /\ 
Lim  z )  ->  x  e.  z )  <->  ( z  e.  On  ->  ( Lim  z  ->  x  e.  z ) ) )
18 con34b 284 . . . . . . . . . 10  |-  ( ( Lim  z  ->  x  e.  z )  <->  ( -.  x  e.  z  ->  -. 
Lim  z ) )
19 ibar 491 . . . . . . . . . . 11  |-  ( z  e.  On  ->  ( -.  Lim  z  <->  ( z  e.  On  /\  -.  Lim  z ) ) )
2019imbi2d 308 . . . . . . . . . 10  |-  ( z  e.  On  ->  (
( -.  x  e.  z  ->  -.  Lim  z
)  <->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2118, 20syl5bb 249 . . . . . . . . 9  |-  ( z  e.  On  ->  (
( Lim  z  ->  x  e.  z )  <->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2221pm5.74i 237 . . . . . . . 8  |-  ( ( z  e.  On  ->  ( Lim  z  ->  x  e.  z ) )  <->  ( z  e.  On  ->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2316, 17, 223bitri 263 . . . . . . 7  |-  ( ( Lim  z  ->  x  e.  z )  <->  ( z  e.  On  ->  ( -.  x  e.  z  ->  ( z  e.  On  /\  -.  Lim  z ) ) ) )
2411, 23syl6rbbr 256 . . . . . 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 434 . . . . . . 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 448 . . . . . . . . 9  |-  ( ( z  e.  On  /\  z  e.  suc  x )  ->  z  e.  suc  x )
27 suceloni 4784 . . . . . . . . . . 11  |-  ( x  e.  On  ->  suc  x  e.  On )
28 onelon 4598 . . . . . . . . . . . 12  |-  ( ( suc  x  e.  On  /\  z  e.  suc  x
)  ->  z  e.  On )
2928ex 424 . . . . . . . . . . 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 538 . . . . . . . . 9  |-  ( x  e.  On  ->  (
z  e.  suc  x  ->  ( z  e.  On  /\  z  e.  suc  x
) ) )
3226, 31impbid2 196 . . . . . . . 8  |-  ( x  e.  On  ->  (
( z  e.  On  /\  z  e.  suc  x
)  <->  z  e.  suc  x ) )
3332imbi1d 309 . . . . . . 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 251 . . . . . 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 245 . . . . 5  |-  ( x  e.  On  ->  (
( Lim  z  ->  x  e.  z )  <->  ( z  e.  suc  x  ->  z  e.  { y  e.  On  |  -.  Lim  y } ) ) )
3635albidv 1635 . . . 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 3329 . . . 4  |-  ( suc  x  C_  { y  e.  On  |  -.  Lim  y }  <->  A. z ( z  e.  suc  x  -> 
z  e.  { y  e.  On  |  -.  Lim  y } ) )
3836, 37syl6bbr 255 . . 3  |-  ( x  e.  On  ->  ( A. z ( Lim  z  ->  x  e.  z )  <->  suc  x  C_  { y  e.  On  |  -.  Lim  y } ) )
3938rabbiia 2938 . 2  |-  { x  e.  On  |  A. z
( Lim  z  ->  x  e.  z ) }  =  { x  e.  On  |  suc  x  C_ 
{ y  e.  On  |  -.  Lim  y } }
401, 39eqtri 2455 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 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    C_ wss 3312   Oncon0 4573   Lim wlim 4574   suc csuc 4575   omcom 4836
This theorem is referenced by:  omsson  4840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837
  Copyright terms: Public domain W3C validator