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Theorem hartogs 7505
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 8430- 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 4598 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
2 vex 2951 . . . . . . . . . . . . 13  |-  z  e. 
_V
3 onelss 4615 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
43imp 419 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  C_  z )
5 ssdomg 7145 . . . . . . . . . . . . 13  |-  ( z  e.  _V  ->  (
y  C_  z  ->  y  ~<_  z ) )
62, 4, 5mpsyl 61 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  ~<_  z )
71, 6jca 519 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  y  e.  z )  ->  ( y  e.  On  /\  y  ~<_  z ) )
8 domtr 7152 . . . . . . . . . . . . 13  |-  ( ( y  ~<_  z  /\  z  ~<_  A )  ->  y  ~<_  A )
98anim2i 553 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  ( y  ~<_  z  /\  z  ~<_  A ) )  ->  ( y  e.  On  /\  y  ~<_  A ) )
109anassrs 630 . . . . . . . . . . 11  |-  ( ( ( y  e.  On  /\  y  ~<_  z )  /\  z  ~<_  A )  -> 
( y  e.  On  /\  y  ~<_  A ) )
117, 10sylan 458 . . . . . . . . . 10  |-  ( ( ( z  e.  On  /\  y  e.  z )  /\  z  ~<_  A )  ->  ( y  e.  On  /\  y  ~<_  A ) )
1211exp31 588 . . . . . . . . 9  |-  ( z  e.  On  ->  (
y  e.  z  -> 
( z  ~<_  A  -> 
( y  e.  On  /\  y  ~<_  A ) ) ) )
1312com12 29 . . . . . . . 8  |-  ( y  e.  z  ->  (
z  e.  On  ->  ( z  ~<_  A  ->  (
y  e.  On  /\  y  ~<_  A ) ) ) )
1413imp3a 421 . . . . . . 7  |-  ( y  e.  z  ->  (
( z  e.  On  /\  z  ~<_  A )  -> 
( y  e.  On  /\  y  ~<_  A ) ) )
15 breq1 4207 . . . . . . . 8  |-  ( x  =  z  ->  (
x  ~<_  A  <->  z  ~<_  A ) )
1615elrab 3084 . . . . . . 7  |-  ( z  e.  { x  e.  On  |  x  ~<_  A }  <->  ( z  e.  On  /\  z  ~<_  A ) )
17 breq1 4207 . . . . . . . 8  |-  ( x  =  y  ->  (
x  ~<_  A  <->  y  ~<_  A ) )
1817elrab 3084 . . . . . . 7  |-  ( y  e.  { x  e.  On  |  x  ~<_  A }  <->  ( y  e.  On  /\  y  ~<_  A ) )
1914, 16, 183imtr4g 262 . . . . . 6  |-  ( y  e.  z  ->  (
z  e.  { x  e.  On  |  x  ~<_  A }  ->  y  e.  { x  e.  On  |  x  ~<_  A } ) )
2019imp 419 . . . . 5  |-  ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<_  A } )  ->  y  e.  { x  e.  On  |  x  ~<_  A }
)
2120gen2 1556 . . . 4  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<_  A }
)  ->  y  e.  { x  e.  On  |  x  ~<_  A } )
22 dftr2 4296 . . . 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 201 . . 3  |-  Tr  {
x  e.  On  |  x  ~<_  A }
24 ssrab2 3420 . . 3  |-  { x  e.  On  |  x  ~<_  A }  C_  On
25 ordon 4755 . . 3  |-  Ord  On
26 trssord 4590 . . 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 1279 . 2  |-  Ord  {
x  e.  On  |  x  ~<_  A }
28 eqid 2435 . . . 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 2435 . . . 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 7504 . . 3  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  _V )
31 elong 4581 . . 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 225 1  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725   E.wrex 2698   {crab 2701   _Vcvv 2948    \ cdif 3309    C_ wss 3312   class class class wbr 4204   {copab 4257   Tr wtr 4294    _E cep 4484    _I cid 4485    We wwe 4532   Ord word 4572   Oncon0 4573    X. cxp 4868   dom cdm 4870    |` cres 4872   ` cfv 5446    ~<_ cdom 7099  OrdIsocoi 7470
This theorem is referenced by:  card2on  7514  harf  7520  harval  7522
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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-riota 6541  df-recs 6625  df-en 7102  df-dom 7103  df-oi 7471
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