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Theorem ondomon 8185
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 7259. (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 4417 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
2 vex 2791 . . . . . . . . . . . . 13  |-  z  e. 
_V
3 onelss 4434 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
43imp 418 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  C_  z )
5 ssdomg 6907 . . . . . . . . . . . . 13  |-  ( z  e.  _V  ->  (
y  C_  z  ->  y  ~<_  z ) )
62, 4, 5mpsyl 59 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  ~<_  z )
71, 6jca 518 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  y  e.  z )  ->  ( y  e.  On  /\  y  ~<_  z ) )
8 domtr 6914 . . . . . . . . . . . . 13  |-  ( ( y  ~<_  z  /\  z  ~<_  A )  ->  y  ~<_  A )
98anim2i 552 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  ( y  ~<_  z  /\  z  ~<_  A ) )  ->  ( y  e.  On  /\  y  ~<_  A ) )
109anassrs 629 . . . . . . . . . . 11  |-  ( ( ( y  e.  On  /\  y  ~<_  z )  /\  z  ~<_  A )  -> 
( y  e.  On  /\  y  ~<_  A ) )
117, 10sylan 457 . . . . . . . . . 10  |-  ( ( ( z  e.  On  /\  y  e.  z )  /\  z  ~<_  A )  ->  ( y  e.  On  /\  y  ~<_  A ) )
1211exp31 587 . . . . . . . . 9  |-  ( z  e.  On  ->  (
y  e.  z  -> 
( z  ~<_  A  -> 
( y  e.  On  /\  y  ~<_  A ) ) ) )
1312com12 27 . . . . . . . 8  |-  ( y  e.  z  ->  (
z  e.  On  ->  ( z  ~<_  A  ->  (
y  e.  On  /\  y  ~<_  A ) ) ) )
1413imp3a 420 . . . . . . 7  |-  ( y  e.  z  ->  (
( z  e.  On  /\  z  ~<_  A )  -> 
( y  e.  On  /\  y  ~<_  A ) ) )
15 breq1 4026 . . . . . . . 8  |-  ( x  =  z  ->  (
x  ~<_  A  <->  z  ~<_  A ) )
1615elrab 2923 . . . . . . 7  |-  ( z  e.  { x  e.  On  |  x  ~<_  A }  <->  ( z  e.  On  /\  z  ~<_  A ) )
17 breq1 4026 . . . . . . . 8  |-  ( x  =  y  ->  (
x  ~<_  A  <->  y  ~<_  A ) )
1817elrab 2923 . . . . . . 7  |-  ( y  e.  { x  e.  On  |  x  ~<_  A }  <->  ( y  e.  On  /\  y  ~<_  A ) )
1914, 16, 183imtr4g 261 . . . . . 6  |-  ( y  e.  z  ->  (
z  e.  { x  e.  On  |  x  ~<_  A }  ->  y  e.  { x  e.  On  |  x  ~<_  A } ) )
2019imp 418 . . . . 5  |-  ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<_  A } )  ->  y  e.  { x  e.  On  |  x  ~<_  A }
)
2120gen2 1534 . . . 4  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<_  A }
)  ->  y  e.  { x  e.  On  |  x  ~<_  A } )
22 dftr2 4115 . . . 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 200 . . 3  |-  Tr  {
x  e.  On  |  x  ~<_  A }
24 ssrab2 3258 . . 3  |-  { x  e.  On  |  x  ~<_  A }  C_  On
25 ordon 4574 . . 3  |-  Ord  On
26 trssord 4409 . . 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 1277 . 2  |-  Ord  {
x  e.  On  |  x  ~<_  A }
28 elex 2796 . . . . . 6  |-  ( A  e.  V  ->  A  e.  _V )
29 canth2g 7015 . . . . . . . . 9  |-  ( A  e.  _V  ->  A  ~<  ~P A )
30 domsdomtr 6996 . . . . . . . . 9  |-  ( ( x  ~<_  A  /\  A  ~<  ~P A )  ->  x  ~<  ~P A )
3129, 30sylan2 460 . . . . . . . 8  |-  ( ( x  ~<_  A  /\  A  e.  _V )  ->  x  ~<  ~P A )
3231expcom 424 . . . . . . 7  |-  ( A  e.  _V  ->  (
x  ~<_  A  ->  x  ~<  ~P A ) )
3332ralrimivw 2627 . . . . . 6  |-  ( A  e.  _V  ->  A. x  e.  On  ( x  ~<_  A  ->  x  ~<  ~P A
) )
3428, 33syl 15 . . . . 5  |-  ( A  e.  V  ->  A. x  e.  On  ( x  ~<_  A  ->  x  ~<  ~P A
) )
35 ss2rab 3249 . . . . 5  |-  ( { x  e.  On  |  x  ~<_  A }  C_  { x  e.  On  |  x  ~<  ~P A }  <->  A. x  e.  On  (
x  ~<_  A  ->  x  ~<  ~P A ) )
3634, 35sylibr 203 . . . 4  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  C_  { x  e.  On  |  x  ~<  ~P A } )
37 pwexg 4194 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
38 numth3 8097 . . . . . 6  |-  ( ~P A  e.  _V  ->  ~P A  e.  dom  card )
39 cardval2 7624 . . . . . 6  |-  ( ~P A  e.  dom  card  -> 
( card `  ~P A )  =  { x  e.  On  |  x  ~<  ~P A } )
4037, 38, 393syl 18 . . . . 5  |-  ( A  e.  V  ->  ( card `  ~P A )  =  { x  e.  On  |  x  ~<  ~P A } )
41 fvex 5539 . . . . 5  |-  ( card `  ~P A )  e. 
_V
4240, 41syl6eqelr 2372 . . . 4  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<  ~P A }  e.  _V )
43 ssexg 4160 . . . 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 642 . . 3  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  _V )
45 elong 4400 . . 3  |-  ( { x  e.  On  |  x  ~<_  A }  e.  _V  ->  ( { x  e.  On  |  x  ~<_  A }  e.  On  <->  Ord  { x  e.  On  |  x  ~<_  A } ) )
4644, 45syl 15 . 2  |-  ( A  e.  V  ->  ( { x  e.  On  |  x  ~<_  A }  e.  On  <->  Ord  { x  e.  On  |  x  ~<_  A } ) )
4727, 46mpbiri 224 1  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023   Tr wtr 4113   Ord word 4391   Oncon0 4392   dom cdm 4689   ` cfv 5255    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-ac2 8089
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-card 7572  df-ac 7743
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