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Theorem harcard 8355
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 7983 . 2  |-  (har `  A )  e.  On
2 harndom 7986 . . . . . . 7  |-  -.  (har `  A )  ~<_  A
3 simpll 753 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  x  e.  On )
4 simpr 461 . . . . . . . . . . 11  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  y  e.  (har `  A ) )
5 elharval 7985 . . . . . . . . . . 11  |-  ( y  e.  (har `  A
)  <->  ( y  e.  On  /\  y  ~<_  A ) )
64, 5sylib 196 . . . . . . . . . 10  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  ( y  e.  On  /\  y  ~<_  A ) )
76simpld 459 . . . . . . . . 9  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  y  e.  On )
8 ontri1 4912 . . . . . . . . 9  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  C_  y  <->  -.  y  e.  x ) )
93, 7, 8syl2anc 661 . . . . . . . 8  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  ( x  C_  y  <->  -.  y  e.  x ) )
10 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( x  e.  On  /\  (har `  A )  ~~  x
)  /\  y  e.  (har `  A ) )  /\  x  C_  y
)  ->  (har `  A
)  ~~  x )
11 vex 3116 . . . . . . . . . . . 12  |-  y  e. 
_V
12 ssdomg 7558 . . . . . . . . . . . 12  |-  ( y  e.  _V  ->  (
x  C_  y  ->  x  ~<_  y ) )
1311, 12ax-mp 5 . . . . . . . . . . 11  |-  ( x 
C_  y  ->  x  ~<_  y )
146simprd 463 . . . . . . . . . . 11  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  y  ~<_  A )
15 domtr 7565 . . . . . . . . . . 11  |-  ( ( x  ~<_  y  /\  y  ~<_  A )  ->  x  ~<_  A )
1613, 14, 15syl2anr 478 . . . . . . . . . 10  |-  ( ( ( ( x  e.  On  /\  (har `  A )  ~~  x
)  /\  y  e.  (har `  A ) )  /\  x  C_  y
)  ->  x  ~<_  A )
17 endomtr 7570 . . . . . . . . . 10  |-  ( ( (har `  A )  ~~  x  /\  x  ~<_  A )  ->  (har `  A )  ~<_  A )
1810, 16, 17syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( x  e.  On  /\  (har `  A )  ~~  x
)  /\  y  e.  (har `  A ) )  /\  x  C_  y
)  ->  (har `  A
)  ~<_  A )
1918ex 434 . . . . . . . 8  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  ( x  C_  y  ->  (har `  A
)  ~<_  A ) )
209, 19sylbird 235 . . . . . . 7  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  ( -.  y  e.  x  ->  (har
`  A )  ~<_  A ) )
212, 20mt3i 126 . . . . . 6  |-  ( ( ( x  e.  On  /\  (har `  A )  ~~  x )  /\  y  e.  (har `  A )
)  ->  y  e.  x )
2221ex 434 . . . . 5  |-  ( ( x  e.  On  /\  (har `  A )  ~~  x )  ->  (
y  e.  (har `  A )  ->  y  e.  x ) )
2322ssrdv 3510 . . . 4  |-  ( ( x  e.  On  /\  (har `  A )  ~~  x )  ->  (har `  A )  C_  x
)
2423ex 434 . . 3  |-  ( x  e.  On  ->  (
(har `  A )  ~~  x  ->  (har `  A )  C_  x
) )
2524rgen 2824 . 2  |-  A. x  e.  On  ( (har `  A )  ~~  x  ->  (har `  A )  C_  x )
26 iscard2 8353 . 2  |-  ( (
card `  (har `  A
) )  =  (har
`  A )  <->  ( (har `  A )  e.  On  /\ 
A. x  e.  On  ( (har `  A )  ~~  x  ->  (har `  A )  C_  x
) ) )
271, 25, 26mpbir2an 918 1  |-  ( card `  (har `  A )
)  =  (har `  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476   class class class wbr 4447   Oncon0 4878   ` cfv 5586    ~~ cen 7510    ~<_ cdom 7511  harchar 7978   cardccrd 8312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-recs 7039  df-er 7308  df-en 7514  df-dom 7515  df-oi 7931  df-har 7980  df-card 8316
This theorem is referenced by:  cardprclem  8356  alephcard  8447  pwcfsdom  8954  hargch  9047
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