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

Theorem ondomon 8955
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 7987. (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 4912 . . . . . . . . . . . 12  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
2 vex 3112 . . . . . . . . . . . . 13  |-  z  e. 
_V
3 onelss 4929 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
43imp 429 . . . . . . . . . . . . 13  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  C_  z )
5 ssdomg 7580 . . . . . . . . . . . . 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 7587 . . . . . . . . . . . . 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 4459 . . . . . . . 8  |-  ( x  =  z  ->  (
x  ~<_  A  <->  z  ~<_  A ) )
1615elrab 3257 . . . . . . 7  |-  ( z  e.  { x  e.  On  |  x  ~<_  A }  <->  ( z  e.  On  /\  z  ~<_  A ) )
17 breq1 4459 . . . . . . . 8  |-  ( x  =  y  ->  (
x  ~<_  A  <->  y  ~<_  A ) )
1817elrab 3257 . . . . . . 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 1620 . . . 4  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  On  |  x  ~<_  A }
)  ->  y  e.  { x  e.  On  |  x  ~<_  A } )
22 dftr2 4552 . . . 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 3581 . . 3  |-  { x  e.  On  |  x  ~<_  A }  C_  On
25 ordon 6617 . . 3  |-  Ord  On
26 trssord 4904 . . 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 3118 . . . . . 6  |-  ( A  e.  V  ->  A  e.  _V )
29 canth2g 7690 . . . . . . . . 9  |-  ( A  e.  _V  ->  A  ~<  ~P A )
30 domsdomtr 7671 . . . . . . . . 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 2872 . . . . . 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 3572 . . . . 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 4640 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
38 numth3 8867 . . . . . 6  |-  ( ~P A  e.  _V  ->  ~P A  e.  dom  card )
39 cardval2 8389 . . . . . 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 5882 . . . . 5  |-  ( card `  ~P A )  e. 
_V
4240, 41syl6eqelr 2554 . . . 4  |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<  ~P A }  e.  _V )
43 ssexg 4602 . . . 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 4895 . . 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 1393    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   class class class wbr 4456   Tr wtr 4550   Ord word 4886   Oncon0 4887   dom cdm 5008   ` cfv 5594    ~<_ cdom 7533    ~< csdm 7534   cardccrd 8333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-ac2 8860
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-recs 7060  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-card 8337  df-ac 8514
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator