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Theorem hartogs 7861
Description: Given any set, the Hartogs number of the set is the least ordinal not dominated by that set. This theorem proves that there is always an ordinal which satisfies this. (This theorem can be proven trivially using the AC - see theorem ondomon 8830- but this proof works in ZF.) (Contributed by Jeff Hankins, 22-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
hartogs  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem hartogs
Dummy variables  g 
r  s  t  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onelon 4844 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
2 vex 3073 . . . . . . . . . . . . 13  |-  z  e. 
_V
3 onelss 4861 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
43imp 429 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  C_  z )
5 ssdomg 7457 . . . . . . . . . . . . 13  |-  ( z  e.  _V  ->  (
y  C_  z  ->  y  ~<_  z ) )
62, 4, 5mpsyl 63 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  ~<_  z )
71, 6jca 532 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  y  e.  z )  ->  ( y  e.  On  /\  y  ~<_  z ) )
8 domtr 7464 . . . . . . . . . . . . 13  |-  ( ( y  ~<_  z  /\  z  ~<_  A )  ->  y  ~<_  A )
98anim2i 569 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  ( y  ~<_  z  /\  z  ~<_  A ) )  ->  ( y  e.  On  /\  y  ~<_  A ) )
109anassrs 648 . . . . . . . . . . 11  |-  ( ( ( y  e.  On  /\  y  ~<_  z )  /\  z  ~<_  A )  -> 
( y  e.  On  /\  y  ~<_  A ) )
117, 10sylan 471 . . . . . . . . . 10  |-  ( ( ( z  e.  On  /\  y  e.  z )  /\  z  ~<_  A )  ->  ( y  e.  On  /\  y  ~<_  A ) )
1211exp31 604 . . . . . . . . 9  |-  ( z  e.  On  ->  (
y  e.  z  -> 
( z  ~<_  A  -> 
( y  e.  On  /\  y  ~<_  A ) ) ) )
1312com12 31 . . . . . . . 8  |-  ( y  e.  z  ->  (
z  e.  On  ->  ( z  ~<_  A  ->  (
y  e.  On  /\  y  ~<_  A ) ) ) )
1413impd 431 . . . . . . 7  |-  ( y  e.  z  ->  (
( z  e.  On  /\  z  ~<_  A )  -> 
( y  e.  On  /\  y  ~<_  A ) ) )
15 breq1 4395 . . . . . . . 8  |-  ( x  =  z  ->  (
x  ~<_  A  <->  z  ~<_  A ) )
1615elrab 3216 . . . . . . 7  |-  ( z  e.  { x  e.  On  |  x  ~<_  A }  <->  ( z  e.  On  /\  z  ~<_  A ) )
17 breq1 4395 . . . . . . . 8  |-  ( x  =  y  ->  (
x  ~<_  A  <->  y  ~<_  A ) )
1817elrab 3216 . . . . . . 7  |-  ( y  e.  { x  e.  On  |  x  ~<_  A }  <->  ( y  e.  On  /\  y  ~<_  A ) )
1914, 16, 183imtr4g 270 . . . . . 6  |-  ( y  e.  z  ->  (
z  e.  { x  e.  On  |  x  ~<_  A }  ->  y  e.  { x  e.  On  |  x  ~<_  A } ) )
2019imp 429 . . . . 5  |-  ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<_  A } )  ->  y  e.  { x  e.  On  |  x  ~<_  A }
)
2120gen2 1593 . . . 4  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<_  A }
)  ->  y  e.  { x  e.  On  |  x  ~<_  A } )
22 dftr2 4487 . . . 4  |-  ( Tr 
{ x  e.  On  |  x  ~<_  A }  <->  A. y A. z ( ( y  e.  z  /\  z  e.  {
x  e.  On  |  x  ~<_  A } )  ->  y  e.  {
x  e.  On  |  x  ~<_  A } ) )
2321, 22mpbir 209 . . 3  |-  Tr  {
x  e.  On  |  x  ~<_  A }
24 ssrab2 3537 . . 3  |-  { x  e.  On  |  x  ~<_  A }  C_  On
25 ordon 6496 . . 3  |-  Ord  On
26 trssord 4836 . . 3  |-  ( ( Tr  { x  e.  On  |  x  ~<_  A }  /\  { x  e.  On  |  x  ~<_  A }  C_  On  /\  Ord  On )  ->  Ord  { x  e.  On  |  x  ~<_  A } )
2723, 24, 25, 26mp3an 1315 . 2  |-  Ord  {
x  e.  On  |  x  ~<_  A }
28 eqid 2451 . . . 4  |-  { <. r ,  y >.  |  ( ( ( dom  r  C_  A  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) }  =  { <. r ,  y >.  |  ( ( ( dom  r  C_  A  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
) )  /\  (
r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) }
29 eqid 2451 . . . 4  |-  { <. s ,  t >.  |  E. w  e.  y  E. z  e.  y  (
( s  =  ( g `  w )  /\  t  =  ( g `  z ) )  /\  w  _E  z ) }  =  { <. s ,  t
>.  |  E. w  e.  y  E. z  e.  y  ( (
s  =  ( g `
 w )  /\  t  =  ( g `  z ) )  /\  w  _E  z ) }
3028, 29hartogslem2 7860 . . 3  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  _V )
31 elong 4827 . . 3  |-  ( { x  e.  On  |  x  ~<_  A }  e.  _V  ->  ( { x  e.  On  |  x  ~<_  A }  e.  On  <->  Ord  { x  e.  On  |  x  ~<_  A } ) )
3230, 31syl 16 . 2  |-  ( A  e.  V  ->  ( { x  e.  On  |  x  ~<_  A }  e.  On  <->  Ord  { x  e.  On  |  x  ~<_  A } ) )
3327, 32mpbiri 233 1  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1368    = wceq 1370    e. wcel 1758   E.wrex 2796   {crab 2799   _Vcvv 3070    \ cdif 3425    C_ wss 3428   class class class wbr 4392   {copab 4449   Tr wtr 4485    _E cep 4730    _I cid 4731    We wwe 4778   Ord word 4818   Oncon0 4819    X. cxp 4938   dom cdm 4940    |` cres 4942   ` cfv 5518    ~<_ cdom 7410  OrdIsocoi 7826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-recs 6934  df-en 7413  df-dom 7414  df-oi 7827
This theorem is referenced by:  card2on  7872  harf  7878  harval  7880
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