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Theorem ondomon 8139
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 7213. (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
StepHypRef Expression
1 onelon 4375 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
2 vex 2760 . . . . . . . . . . . . 13  |-  z  e. 
_V
3 onelss 4392 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
43imp 420 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  C_  z )
5 ssdomg 6861 . . . . . . . . . . . . 13  |-  ( z  e.  _V  ->  (
y  C_  z  ->  y  ~<_  z ) )
62, 4, 5mpsyl 61 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  ~<_  z )
71, 6jca 520 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  y  e.  z )  ->  ( y  e.  On  /\  y  ~<_  z ) )
8 domtr 6868 . . . . . . . . . . . . 13  |-  ( ( y  ~<_  z  /\  z  ~<_  A )  ->  y  ~<_  A )
98anim2i 555 . . . . . . . . . . . 12  |-  ( ( y  e.  On  /\  ( y  ~<_  z  /\  z  ~<_  A ) )  ->  ( y  e.  On  /\  y  ~<_  A ) )
109anassrs 632 . . . . . . . . . . 11  |-  ( ( ( y  e.  On  /\  y  ~<_  z )  /\  z  ~<_  A )  -> 
( y  e.  On  /\  y  ~<_  A ) )
117, 10sylan 459 . . . . . . . . . 10  |-  ( ( ( z  e.  On  /\  y  e.  z )  /\  z  ~<_  A )  ->  ( y  e.  On  /\  y  ~<_  A ) )
1211exp31 590 . . . . . . . . 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 422 . . . . . . 7  |-  ( y  e.  z  ->  (
( z  e.  On  /\  z  ~<_  A )  -> 
( y  e.  On  /\  y  ~<_  A ) ) )
15 breq1 3986 . . . . . . . 8  |-  ( x  =  z  ->  (
x  ~<_  A  <->  z  ~<_  A ) )
1615elrab 2891 . . . . . . 7  |-  ( z  e.  { x  e.  On  |  x  ~<_  A }  <->  ( z  e.  On  /\  z  ~<_  A ) )
17 breq1 3986 . . . . . . . 8  |-  ( x  =  y  ->  (
x  ~<_  A  <->  y  ~<_  A ) )
1817elrab 2891 . . . . . . 7  |-  ( y  e.  { x  e.  On  |  x  ~<_  A }  <->  ( y  e.  On  /\  y  ~<_  A ) )
1914, 16, 183imtr4g 263 . . . . . 6  |-  ( y  e.  z  ->  (
z  e.  { x  e.  On  |  x  ~<_  A }  ->  y  e.  { x  e.  On  |  x  ~<_  A } ) )
2019imp 420 . . . . 5  |-  ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<_  A } )  ->  y  e.  { x  e.  On  |  x  ~<_  A }
)
2120gen2 1541 . . . 4  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<_  A }
)  ->  y  e.  { x  e.  On  |  x  ~<_  A } )
22 dftr2 4075 . . . 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 202 . . 3  |-  Tr  {
x  e.  On  |  x  ~<_  A }
24 ssrab2 3219 . . 3  |-  { x  e.  On  |  x  ~<_  A }  C_  On
25 ordon 4532 . . 3  |-  Ord  On
26 trssord 4367 . . 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 1282 . 2  |-  Ord  {
x  e.  On  |  x  ~<_  A }
28 elex 2765 . . . . . 6  |-  ( A  e.  V  ->  A  e.  _V )
29 canth2g 6969 . . . . . . . . 9  |-  ( A  e.  _V  ->  A  ~<  ~P A )
30 domsdomtr 6950 . . . . . . . . 9  |-  ( ( x  ~<_  A  /\  A  ~<  ~P A )  ->  x  ~<  ~P A )
3129, 30sylan2 462 . . . . . . . 8  |-  ( ( x  ~<_  A  /\  A  e.  _V )  ->  x  ~<  ~P A )
3231expcom 426 . . . . . . 7  |-  ( A  e.  _V  ->  (
x  ~<_  A  ->  x  ~<  ~P A ) )
3332ralrimivw 2600 . . . . . 6  |-  ( A  e.  _V  ->  A. x  e.  On  ( x  ~<_  A  ->  x  ~<  ~P A
) )
3428, 33syl 17 . . . . 5  |-  ( A  e.  V  ->  A. x  e.  On  ( x  ~<_  A  ->  x  ~<  ~P A
) )
35 ss2rab 3210 . . . . 5  |-  ( { x  e.  On  |  x  ~<_  A }  C_  { x  e.  On  |  x  ~<  ~P A }  <->  A. x  e.  On  (
x  ~<_  A  ->  x  ~<  ~P A ) )
3634, 35sylibr 205 . . . 4  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  C_  { x  e.  On  |  x  ~<  ~P A } )
37 pwexg 4152 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
38 numth3 8051 . . . . . 6  |-  ( ~P A  e.  _V  ->  ~P A  e.  dom  card )
39 cardval2 7578 . . . . . 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 5458 . . . . 5  |-  ( card `  ~P A )  e. 
_V
4240, 41syl6eqelr 2345 . . . 4  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<  ~P A }  e.  _V )
43 ssexg 4120 . . . 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 645 . . 3  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  _V )
45 elong 4358 . . 3  |-  ( { x  e.  On  |  x  ~<_  A }  e.  _V  ->  ( { x  e.  On  |  x  ~<_  A }  e.  On  <->  Ord  { x  e.  On  |  x  ~<_  A } ) )
4644, 45syl 17 . 2  |-  ( A  e.  V  ->  ( { x  e.  On  |  x  ~<_  A }  e.  On  <->  Ord  { x  e.  On  |  x  ~<_  A } ) )
4727, 46mpbiri 226 1  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532    = wceq 1619    e. wcel 1621   A.wral 2516   {crab 2520   _Vcvv 2757    C_ wss 3113   ~Pcpw 3585   class class class wbr 3983   Tr wtr 4073   Ord word 4349   Oncon0 4350   dom cdm 4647   ` cfv 4659    ~<_ cdom 6815    ~< csdm 6816   cardccrd 7522
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-ac2 8043
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-suc 4356  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-iota 6211  df-riota 6258  df-recs 6342  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-card 7526  df-ac 7697
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