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Theorem cardsdomel 8367
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 8346 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
21ensymd 7578 . . . . . 6  |-  ( B  e.  dom  card  ->  B 
~~  ( card `  B
) )
3 sdomentr 7663 . . . . . 6  |-  ( ( A  ~<  B  /\  B  ~~  ( card `  B
) )  ->  A  ~<  ( card `  B
) )
42, 3sylan2 474 . . . . 5  |-  ( ( A  ~<  B  /\  B  e.  dom  card )  ->  A  ~<  ( card `  B ) )
5 ssdomg 7573 . . . . . . . 8  |-  ( A  e.  On  ->  (
( card `  B )  C_  A  ->  ( card `  B )  ~<_  A ) )
6 cardon 8337 . . . . . . . . 9  |-  ( card `  B )  e.  On
7 domtriord 7675 . . . . . . . . 9  |-  ( ( ( card `  B
)  e.  On  /\  A  e.  On )  ->  ( ( card `  B
)  ~<_  A  <->  -.  A  ~<  ( card `  B
) ) )
86, 7mpan 670 . . . . . . . 8  |-  ( A  e.  On  ->  (
( card `  B )  ~<_  A 
<->  -.  A  ~<  ( card `  B ) ) )
95, 8sylibd 214 . . . . . . 7  |-  ( A  e.  On  ->  (
( card `  B )  C_  A  ->  -.  A  ~<  ( card `  B
) ) )
109con2d 115 . . . . . 6  |-  ( A  e.  On  ->  ( A  ~<  ( card `  B
)  ->  -.  ( card `  B )  C_  A ) )
11 ontri1 4918 . . . . . . . 8  |-  ( ( ( card `  B
)  e.  On  /\  A  e.  On )  ->  ( ( card `  B
)  C_  A  <->  -.  A  e.  ( card `  B
) ) )
126, 11mpan 670 . . . . . . 7  |-  ( A  e.  On  ->  (
( card `  B )  C_  A  <->  -.  A  e.  ( card `  B )
) )
1312con2bid 329 . . . . . 6  |-  ( A  e.  On  ->  ( A  e.  ( card `  B )  <->  -.  ( card `  B )  C_  A ) )
1410, 13sylibrd 234 . . . . 5  |-  ( A  e.  On  ->  ( A  ~<  ( card `  B
)  ->  A  e.  ( card `  B )
) )
154, 14syl5 32 . . . 4  |-  ( A  e.  On  ->  (
( A  ~<  B  /\  B  e.  dom  card )  ->  A  e.  ( card `  B ) ) )
1615expcomd 438 . . 3  |-  ( A  e.  On  ->  ( B  e.  dom  card  ->  ( A  ~<  B  ->  A  e.  ( card `  B
) ) ) )
1716imp 429 . 2  |-  ( ( A  e.  On  /\  B  e.  dom  card )  ->  ( A  ~<  B  ->  A  e.  ( card `  B ) ) )
18 cardsdomelir 8366 . 2  |-  ( A  e.  ( card `  B
)  ->  A  ~<  B )
1917, 18impbid1 203 1  |-  ( ( A  e.  On  /\  B  e.  dom  card )  ->  ( A  ~<  B  <->  A  e.  ( card `  B )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767    C_ wss 3481   class class class wbr 4453   Oncon0 4884   dom cdm 5005   ` cfv 5594    ~~ cen 7525    ~<_ cdom 7526    ~< csdm 7527   cardccrd 8328
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-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-card 8332
This theorem is referenced by:  iscard  8368  cardval2  8384  infxpenlem  8403  alephnbtwn  8464  alephnbtwn2  8465  alephord2  8469  alephsdom  8479  pwsdompw  8596  inaprc  9226
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