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Theorem cardsdomel 7603
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 4-Jun-2015.)
Assertion
Ref Expression
cardsdomel  |-  ( ( A  e.  On  /\  B  e.  dom  card )  ->  ( A  ~<  B  <->  A  e.  ( card `  B )
) )

Proof of Theorem cardsdomel
StepHypRef Expression
1 cardid2 7582 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
2 ensym 6906 . . . . . . 7  |-  ( (
card `  B )  ~~  B  ->  B  ~~  ( card `  B )
)
31, 2syl 17 . . . . . 6  |-  ( B  e.  dom  card  ->  B 
~~  ( card `  B
) )
4 sdomentr 6991 . . . . . 6  |-  ( ( A  ~<  B  /\  B  ~~  ( card `  B
) )  ->  A  ~<  ( card `  B
) )
53, 4sylan2 462 . . . . 5  |-  ( ( A  ~<  B  /\  B  e.  dom  card )  ->  A  ~<  ( card `  B ) )
6 ssdomg 6903 . . . . . . . 8  |-  ( A  e.  On  ->  (
( card `  B )  C_  A  ->  ( card `  B )  ~<_  A ) )
7 cardon 7573 . . . . . . . . 9  |-  ( card `  B )  e.  On
8 domtriord 7003 . . . . . . . . 9  |-  ( ( ( card `  B
)  e.  On  /\  A  e.  On )  ->  ( ( card `  B
)  ~<_  A  <->  -.  A  ~<  ( card `  B
) ) )
97, 8mpan 653 . . . . . . . 8  |-  ( A  e.  On  ->  (
( card `  B )  ~<_  A 
<->  -.  A  ~<  ( card `  B ) ) )
106, 9sylibd 207 . . . . . . 7  |-  ( A  e.  On  ->  (
( card `  B )  C_  A  ->  -.  A  ~<  ( card `  B
) ) )
1110con2d 109 . . . . . 6  |-  ( A  e.  On  ->  ( A  ~<  ( card `  B
)  ->  -.  ( card `  B )  C_  A ) )
12 ontri1 4426 . . . . . . . 8  |-  ( ( ( card `  B
)  e.  On  /\  A  e.  On )  ->  ( ( card `  B
)  C_  A  <->  -.  A  e.  ( card `  B
) ) )
137, 12mpan 653 . . . . . . 7  |-  ( A  e.  On  ->  (
( card `  B )  C_  A  <->  -.  A  e.  ( card `  B )
) )
1413con2bid 321 . . . . . 6  |-  ( A  e.  On  ->  ( A  e.  ( card `  B )  <->  -.  ( card `  B )  C_  A ) )
1511, 14sylibrd 227 . . . . 5  |-  ( A  e.  On  ->  ( A  ~<  ( card `  B
)  ->  A  e.  ( card `  B )
) )
165, 15syl5 30 . . . 4  |-  ( A  e.  On  ->  (
( A  ~<  B  /\  B  e.  dom  card )  ->  A  e.  ( card `  B ) ) )
1716exp3acom23 1364 . . 3  |-  ( A  e.  On  ->  ( B  e.  dom  card  ->  ( A  ~<  B  ->  A  e.  ( card `  B
) ) ) )
1817imp 420 . 2  |-  ( ( A  e.  On  /\  B  e.  dom  card )  ->  ( A  ~<  B  ->  A  e.  ( card `  B ) ) )
19 cardsdomelir 7602 . 2  |-  ( A  e.  ( card `  B
)  ->  A  ~<  B )
2018, 19impbid1 196 1  |-  ( ( A  e.  On  /\  B  e.  dom  card )  ->  ( A  ~<  B  <->  A  e.  ( card `  B )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1685    C_ wss 3154   class class class wbr 4025   Oncon0 4392   dom cdm 4689   ` cfv 5222    ~~ cen 6856    ~<_ cdom 6857    ~< csdm 6858   cardccrd 7564
This theorem is referenced by:  iscard  7604  cardval2  7620  infxpenlem  7637  alephnbtwn  7694  alephnbtwn2  7695  alephord2  7699  alephsdom  7709  pwsdompw  7826  inaprc  8454
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-card 7568
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