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Theorem harcard 7579
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 7243 . 2  |-  (har `  A )  e.  On
2 harndom 7246 . . . . . . 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 7245 . . . . . . . . . . 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 4398 . . . . . . . . 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 2766 . . . . . . . . . . . 12  |-  y  e. 
_V
12 ssdomg 6875 . . . . . . . . . . . 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 6882 . . . . . . . . . . 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 6887 . . . . . . . . . 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 3160 . . . 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 2583 . 2  |-  A. x  e.  On  ( (har `  A )  ~~  x  ->  (har `  A )  C_  x )
26 iscard2 7577 . 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 2518   _Vcvv 2763    C_ wss 3127   class class class wbr 3997   Oncon0 4364   ` cfv 4673    ~~ cen 6828    ~<_ cdom 6829  harchar 7238   cardccrd 7536
This theorem is referenced by:  cardprclem  7580  alephcard  7665  pwcfsdom  8173  hargch  8267
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-iota 6225  df-riota 6272  df-recs 6356  df-er 6628  df-en 6832  df-dom 6833  df-oi 7193  df-har 7240  df-card 7540
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