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Theorem harcard 7495
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 7159 . 2  |-  (har `  A )  e.  On
2 harndom 7162 . . . . . . 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 7161 . . . . . . . . . . 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 4319 . . . . . . . . 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 2730 . . . . . . . . . . . 12  |-  y  e. 
_V
12 ssdomg 6793 . . . . . . . . . . . 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 6799 . . . . . . . . . . 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 6804 . . . . . . . . . 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 3106 . . . 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 2570 . 2  |-  A. x  e.  On  ( (har `  A )  ~~  x  ->  (har `  A )  C_  x )
26 iscard2 7493 . 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 2509   _Vcvv 2727    C_ wss 3078   class class class wbr 3920   Oncon0 4285   ` cfv 4592    ~~ cen 6746    ~<_ cdom 6747  harchar 7154   cardccrd 7452
This theorem is referenced by:  cardprclem  7496  alephcard  7581  pwcfsdom  8085  hargch  8179
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
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-lim 4290  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-oi 7109  df-har 7156  df-card 7456
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