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Theorem hartogs 7447
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 8372- 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 4548 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
2 vex 2903 . . . . . . . . . . . . 13  |-  z  e. 
_V
3 onelss 4565 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
43imp 419 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  C_  z )
5 ssdomg 7090 . . . . . . . . . . . . 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 7097 . . . . . . . . . . . . 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 4157 . . . . . . . 8  |-  ( x  =  z  ->  (
x  ~<_  A  <->  z  ~<_  A ) )
1615elrab 3036 . . . . . . 7  |-  ( z  e.  { x  e.  On  |  x  ~<_  A }  <->  ( z  e.  On  /\  z  ~<_  A ) )
17 breq1 4157 . . . . . . . 8  |-  ( x  =  y  ->  (
x  ~<_  A  <->  y  ~<_  A ) )
1817elrab 3036 . . . . . . 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 1553 . . . 4  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<_  A }
)  ->  y  e.  { x  e.  On  |  x  ~<_  A } )
22 dftr2 4246 . . . 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 3372 . . 3  |-  { x  e.  On  |  x  ~<_  A }  C_  On
25 ordon 4704 . . 3  |-  Ord  On
26 trssord 4540 . . 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 2388 . . . 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 2388 . . . 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 7446 . . 3  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  _V )
31 elong 4531 . . 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 1546    = wceq 1649    e. wcel 1717   E.wrex 2651   {crab 2654   _Vcvv 2900    \ cdif 3261    C_ wss 3264   class class class wbr 4154   {copab 4207   Tr wtr 4244    _E cep 4434    _I cid 4435    We wwe 4482   Ord word 4522   Oncon0 4523    X. cxp 4817   dom cdm 4819    |` cres 4821   ` cfv 5395    ~<_ cdom 7044  OrdIsocoi 7412
This theorem is referenced by:  card2on  7456  harf  7462  harval  7464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-riota 6486  df-recs 6570  df-en 7047  df-dom 7048  df-oi 7413
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