MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ondomon Unicode version

Theorem ondomon 8067
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 7143. (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 4310 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
2 vex 2730 . . . . . . . . . . . . 13  |-  z  e. 
_V
3 onelss 4327 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
43imp 420 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  C_  z )
5 ssdomg 6793 . . . . . . . . . . . . 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 6799 . . . . . . . . . . . . 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 3923 . . . . . . . 8  |-  ( x  =  z  ->  (
x  ~<_  A  <->  z  ~<_  A ) )
1615elrab 2860 . . . . . . 7  |-  ( z  e.  { x  e.  On  |  x  ~<_  A }  <->  ( z  e.  On  /\  z  ~<_  A ) )
17 breq1 3923 . . . . . . . 8  |-  ( x  =  y  ->  (
x  ~<_  A  <->  y  ~<_  A ) )
1817elrab 2860 . . . . . . 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 4012 . . . 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 3179 . . 3  |-  { x  e.  On  |  x  ~<_  A }  C_  On
25 ordon 4465 . . 3  |-  Ord  On
26 trssord 4302 . . 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 2735 . . . . . 6  |-  ( A  e.  V  ->  A  e.  _V )
29 canth2g 6900 . . . . . . . . 9  |-  ( A  e.  _V  ->  A  ~<  ~P A )
30 domsdomtr 6881 . . . . . . . . 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 2589 . . . . . 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 3170 . . . . 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 4088 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
38 numth3 7981 . . . . . 6  |-  ( ~P A  e.  _V  ->  ~P A  e.  dom  card )
39 cardval2 7508 . . . . . 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 5391 . . . . 5  |-  ( card `  ~P A )  e. 
_V
4240, 41syl6eqelr 2342 . . . 4  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<  ~P A }  e.  _V )
43 ssexg 4057 . . . 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 4293 . . 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 2509   {crab 2512   _Vcvv 2727    C_ wss 3078   ~Pcpw 3530   class class class wbr 3920   Tr wtr 4010   Ord word 4284   Oncon0 4285   dom cdm 4580   ` cfv 4592    ~<_ cdom 6747    ~< csdm 6748   cardccrd 7452
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-ac2 7973
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-iota 6143  df-riota 6190  df-recs 6274  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-card 7456  df-ac 7627
  Copyright terms: Public domain W3C validator