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

Theorem harcard 7544
Description: The class of ordinal numbers dominated by a set is a cardinal number. Theorem 59 of [Suppes] p. 228. (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
harcard  |-  ( card `  (har `  A )
)  =  (har `  A )

Proof of Theorem harcard
StepHypRef Expression
1 harcl 7208 . 2  |-  (har `  A )  e.  On
2 harndom 7211 . . . . . . 7  |-  -.  (har `  A )  ~<_  A
3 simpll 733 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  x  e.  On )
4 simpr 449 . . . . . . . . . . 11  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  y  e.  (har `  A ) )
5 elharval 7210 . . . . . . . . . . 11  |-  ( y  e.  (har `  A
)  <->  ( y  e.  On  /\  y  ~<_  A ) )
64, 5sylib 190 . . . . . . . . . 10  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  ( y  e.  On  /\  y  ~<_  A ) )
76simpld 447 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  y  e.  On )
8 ontri1 4363 . . . . . . . . 9  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  C_  y  <->  -.  y  e.  x ) )
93, 7, 8syl2anc 645 . . . . . . . 8  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  ( x  C_  y  <->  -.  y  e.  x ) )
10 simpllr 738 . . . . . . . . . 10  |-  ( ( ( ( x  e.  On  /\  (har `  A )  ~~  x
)  /\  y  e.  (har `  A ) )  /\  x  C_  y
)  ->  (har `  A
)  ~~  x )
11 vex 2743 . . . . . . . . . . . 12  |-  y  e. 
_V
12 ssdomg 6840 . . . . . . . . . . . 12  |-  ( y  e.  _V  ->  (
x  C_  y  ->  x  ~<_  y ) )
1311, 12ax-mp 10 . . . . . . . . . . 11  |-  ( x 
C_  y  ->  x  ~<_  y )
146simprd 451 . . . . . . . . . . 11  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  y  ~<_  A )
15 domtr 6847 . . . . . . . . . . 11  |-  ( ( x  ~<_  y  /\  y  ~<_  A )  ->  x  ~<_  A )
1613, 14, 15syl2anr 466 . . . . . . . . . 10  |-  ( ( ( ( x  e.  On  /\  (har `  A )  ~~  x
)  /\  y  e.  (har `  A ) )  /\  x  C_  y
)  ->  x  ~<_  A )
17 endomtr 6852 . . . . . . . . . 10  |-  ( ( (har `  A )  ~~  x  /\  x  ~<_  A )  ->  (har `  A )  ~<_  A )
1810, 16, 17syl2anc 645 . . . . . . . . 9  |-  ( ( ( ( x  e.  On  /\  (har `  A )  ~~  x
)  /\  y  e.  (har `  A ) )  /\  x  C_  y
)  ->  (har `  A
)  ~<_  A )
1918ex 425 . . . . . . . 8  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  ( x  C_  y  ->  (har `  A
)  ~<_  A ) )
209, 19sylbird 228 . . . . . . 7  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  ( -.  y  e.  x  ->  (har
`  A )  ~<_  A ) )
212, 20mt3i 120 . . . . . 6  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  y  e.  x )
2221ex 425 . . . . 5  |-  ( ( x  e.  On  /\  (har `  A )  ~~  x )  ->  (
y  e.  (har `  A )  ->  y  e.  x ) )
2322ssrdv 3127 . . . 4  |-  ( ( x  e.  On  /\  (har `  A )  ~~  x )  ->  (har `  A )  C_  x
)
2423ex 425 . . 3  |-  ( x  e.  On  ->  (
(har `  A )  ~~  x  ->  (har `  A )  C_  x
) )
2524rgen 2579 . 2  |-  A. x  e.  On  ( (har `  A )  ~~  x  ->  (har `  A )  C_  x )
26 iscard2 7542 . 2  |-  ( (
card `  (har `  A
) )  =  (har
`  A )  <->  ( (har `  A )  e.  On  /\ 
A. x  e.  On  ( (har `  A )  ~~  x  ->  (har `  A )  C_  x
) ) )
271, 25, 26mpbir2an 891 1  |-  ( card `  (har `  A )
)  =  (har `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   _Vcvv 2740    C_ wss 3094   class class class wbr 3963   Oncon0 4329   ` cfv 4638    ~~ cen 6793    ~<_ cdom 6794  harchar 7203   cardccrd 7501
This theorem is referenced by:  cardprclem  7545  alephcard  7630  pwcfsdom  8138  hargch  8232
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 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-iota 6190  df-riota 6237  df-recs 6321  df-er 6593  df-en 6797  df-dom 6798  df-oi 7158  df-har 7205  df-card 7505
  Copyright terms: Public domain W3C validator