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Theorem ondomon 8934
Description: The collection of ordinal numbers dominated by a set is an ordinal number. (In general, not all collections of ordinal numbers are ordinal.) Theorem 56 of [Suppes] p. 227. This theorem can be proved (with a longer proof) without the Axiom of Choice; see hartogs 7965. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.)
Assertion
Ref Expression
ondomon  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem ondomon
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onelon 4903 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
2 vex 3116 . . . . . . . . . . . . 13  |-  z  e. 
_V
3 onelss 4920 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
43imp 429 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  C_  z )
5 ssdomg 7558 . . . . . . . . . . . . 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 7565 . . . . . . . . . . . . 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 4450 . . . . . . . 8  |-  ( x  =  z  ->  (
x  ~<_  A  <->  z  ~<_  A ) )
1615elrab 3261 . . . . . . 7  |-  ( z  e.  { x  e.  On  |  x  ~<_  A }  <->  ( z  e.  On  /\  z  ~<_  A ) )
17 breq1 4450 . . . . . . . 8  |-  ( x  =  y  ->  (
x  ~<_  A  <->  y  ~<_  A ) )
1817elrab 3261 . . . . . . 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 1602 . . . 4  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<_  A }
)  ->  y  e.  { x  e.  On  |  x  ~<_  A } )
22 dftr2 4542 . . . 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 3585 . . 3  |-  { x  e.  On  |  x  ~<_  A }  C_  On
25 ordon 6596 . . 3  |-  Ord  On
26 trssord 4895 . . 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 1324 . 2  |-  Ord  {
x  e.  On  |  x  ~<_  A }
28 elex 3122 . . . . . 6  |-  ( A  e.  V  ->  A  e.  _V )
29 canth2g 7668 . . . . . . . . 9  |-  ( A  e.  _V  ->  A  ~<  ~P A )
30 domsdomtr 7649 . . . . . . . . 9  |-  ( ( x  ~<_  A  /\  A  ~<  ~P A )  ->  x  ~<  ~P A )
3129, 30sylan2 474 . . . . . . . 8  |-  ( ( x  ~<_  A  /\  A  e.  _V )  ->  x  ~<  ~P A )
3231expcom 435 . . . . . . 7  |-  ( A  e.  _V  ->  (
x  ~<_  A  ->  x  ~<  ~P A ) )
3332ralrimivw 2879 . . . . . 6  |-  ( A  e.  _V  ->  A. x  e.  On  ( x  ~<_  A  ->  x  ~<  ~P A
) )
3428, 33syl 16 . . . . 5  |-  ( A  e.  V  ->  A. x  e.  On  ( x  ~<_  A  ->  x  ~<  ~P A
) )
35 ss2rab 3576 . . . . 5  |-  ( { x  e.  On  |  x  ~<_  A }  C_  { x  e.  On  |  x  ~<  ~P A }  <->  A. x  e.  On  (
x  ~<_  A  ->  x  ~<  ~P A ) )
3634, 35sylibr 212 . . . 4  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  C_  { x  e.  On  |  x  ~<  ~P A } )
37 pwexg 4631 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
38 numth3 8846 . . . . . 6  |-  ( ~P A  e.  _V  ->  ~P A  e.  dom  card )
39 cardval2 8368 . . . . . 6  |-  ( ~P A  e.  dom  card  -> 
( card `  ~P A )  =  { x  e.  On  |  x  ~<  ~P A } )
4037, 38, 393syl 20 . . . . 5  |-  ( A  e.  V  ->  ( card `  ~P A )  =  { x  e.  On  |  x  ~<  ~P A } )
41 fvex 5874 . . . . 5  |-  ( card `  ~P A )  e. 
_V
4240, 41syl6eqelr 2564 . . . 4  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<  ~P A }  e.  _V )
43 ssexg 4593 . . . 4  |-  ( ( { x  e.  On  |  x  ~<_  A }  C_ 
{ x  e.  On  |  x  ~<  ~P A }  /\  { x  e.  On  |  x  ~<  ~P A }  e.  _V )  ->  { x  e.  On  |  x  ~<_  A }  e.  _V )
4436, 42, 43syl2anc 661 . . 3  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  _V )
45 elong 4886 . . 3  |-  ( { x  e.  On  |  x  ~<_  A }  e.  _V  ->  ( { x  e.  On  |  x  ~<_  A }  e.  On  <->  Ord  { x  e.  On  |  x  ~<_  A } ) )
4644, 45syl 16 . 2  |-  ( A  e.  V  ->  ( { x  e.  On  |  x  ~<_  A }  e.  On  <->  Ord  { x  e.  On  |  x  ~<_  A } ) )
4727, 46mpbiri 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   A.wal 1377    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010   class class class wbr 4447   Tr wtr 4540   Ord word 4877   Oncon0 4878   dom cdm 4999   ` cfv 5586    ~<_ cdom 7511    ~< csdm 7512   cardccrd 8312
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-ac2 8839
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-recs 7039  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-card 8316  df-ac 8493
This theorem is referenced by: (None)
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