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Theorem harcard 7607
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 harcl 7271 . 2  |-  (har `  A )  e.  On
2 harndom 7274 . . . . . . 7  |-  -.  (har `  A )  ~<_  A
3 simpll 730 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  x  e.  On )
4 simpr 447 . . . . . . . . . . 11  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  y  e.  (har `  A ) )
5 elharval 7273 . . . . . . . . . . 11  |-  ( y  e.  (har `  A
)  <->  ( y  e.  On  /\  y  ~<_  A ) )
64, 5sylib 188 . . . . . . . . . 10  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  ( y  e.  On  /\  y  ~<_  A ) )
76simpld 445 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  y  e.  On )
8 ontri1 4425 . . . . . . . . 9  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  C_  y  <->  -.  y  e.  x ) )
93, 7, 8syl2anc 642 . . . . . . . 8  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  ( x  C_  y  <->  -.  y  e.  x ) )
10 simpllr 735 . . . . . . . . . 10  |-  ( ( ( ( x  e.  On  /\  (har `  A )  ~~  x
)  /\  y  e.  (har `  A ) )  /\  x  C_  y
)  ->  (har `  A
)  ~~  x )
11 vex 2792 . . . . . . . . . . . 12  |-  y  e. 
_V
12 ssdomg 6903 . . . . . . . . . . . 12  |-  ( y  e.  _V  ->  (
x  C_  y  ->  x  ~<_  y ) )
1311, 12ax-mp 8 . . . . . . . . . . 11  |-  ( x 
C_  y  ->  x  ~<_  y )
146simprd 449 . . . . . . . . . . 11  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  y  ~<_  A )
15 domtr 6910 . . . . . . . . . . 11  |-  ( ( x  ~<_  y  /\  y  ~<_  A )  ->  x  ~<_  A )
1613, 14, 15syl2anr 464 . . . . . . . . . 10  |-  ( ( ( ( x  e.  On  /\  (har `  A )  ~~  x
)  /\  y  e.  (har `  A ) )  /\  x  C_  y
)  ->  x  ~<_  A )
17 endomtr 6915 . . . . . . . . . 10  |-  ( ( (har `  A )  ~~  x  /\  x  ~<_  A )  ->  (har `  A )  ~<_  A )
1810, 16, 17syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( x  e.  On  /\  (har `  A )  ~~  x
)  /\  y  e.  (har `  A ) )  /\  x  C_  y
)  ->  (har `  A
)  ~<_  A )
1918ex 423 . . . . . . . 8  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  ( x  C_  y  ->  (har `  A
)  ~<_  A ) )
209, 19sylbird 226 . . . . . . 7  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  ( -.  y  e.  x  ->  (har
`  A )  ~<_  A ) )
212, 20mt3i 118 . . . . . 6  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  y  e.  x )
2221ex 423 . . . . 5  |-  ( ( x  e.  On  /\  (har `  A )  ~~  x )  ->  (
y  e.  (har `  A )  ->  y  e.  x ) )
2322ssrdv 3186 . . . 4  |-  ( ( x  e.  On  /\  (har `  A )  ~~  x )  ->  (har `  A )  C_  x
)
2423ex 423 . . 3  |-  ( x  e.  On  ->  (
(har `  A )  ~~  x  ->  (har `  A )  C_  x
) )
2524rgen 2609 . 2  |-  A. x  e.  On  ( (har `  A )  ~~  x  ->  (har `  A )  C_  x )
26 iscard2 7605 . 2  |-  ( (
card `  (har `  A
) )  =  (har
`  A )  <->  ( (har `  A )  e.  On  /\ 
A. x  e.  On  ( (har `  A )  ~~  x  ->  (har `  A )  C_  x
) ) )
271, 25, 26mpbir2an 886 1  |-  ( card `  (har `  A )
)  =  (har `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685   A.wral 2544   _Vcvv 2789    C_ wss 3153   class class class wbr 4024   Oncon0 4391   ` cfv 5221    ~~ cen 6856    ~<_ cdom 6857  harchar 7266   cardccrd 7564
This theorem is referenced by:  cardprclem  7608  alephcard  7693  pwcfsdom  8201  hargch  8295
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-iota 6253  df-riota 6300  df-recs 6384  df-er 6656  df-en 6860  df-dom 6861  df-oi 7221  df-har 7268  df-card 7568
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