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Theorem cardsdomel 7491
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 7470 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
2 ensym 6796 . . . . . . 7  |-  ( (
card `  B )  ~~  B  ->  B  ~~  ( card `  B )
)
31, 2syl 17 . . . . . 6  |-  ( B  e.  dom  card  ->  B 
~~  ( card `  B
) )
4 sdomentr 6880 . . . . . 6  |-  ( ( A  ~<  B  /\  B  ~~  ( card `  B
) )  ->  A  ~<  ( card `  B
) )
53, 4sylan2 462 . . . . 5  |-  ( ( A  ~<  B  /\  B  e.  dom  card )  ->  A  ~<  ( card `  B ) )
6 ssdomg 6793 . . . . . . . 8  |-  ( A  e.  On  ->  (
( card `  B )  C_  A  ->  ( card `  B )  ~<_  A ) )
7 cardon 7461 . . . . . . . . 9  |-  ( card `  B )  e.  On
8 domtriord 6892 . . . . . . . . 9  |-  ( ( ( card `  B
)  e.  On  /\  A  e.  On )  ->  ( ( card `  B
)  ~<_  A  <->  -.  A  ~<  ( card `  B
) ) )
97, 8mpan 654 . . . . . . . 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 4319 . . . . . . . 8  |-  ( ( ( card `  B
)  e.  On  /\  A  e.  On )  ->  ( ( card `  B
)  C_  A  <->  -.  A  e.  ( card `  B
) ) )
137, 12mpan 654 . . . . . . 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 1368 . . 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 7490 . 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 1621    C_ wss 3078   class class class wbr 3920   Oncon0 4285   dom cdm 4580   ` cfv 4592    ~~ cen 6746    ~<_ cdom 6747    ~< csdm 6748   cardccrd 7452
This theorem is referenced by:  iscard  7492  cardval2  7508  infxpenlem  7525  alephnbtwn  7582  alephnbtwn2  7583  alephord2  7587  alephsdom  7597  pwsdompw  7714  inaprc  8338
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-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-rab 2516  df-v 2729  df-sbc 2922  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-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-we 4247  df-ord 4288  df-on 4289  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-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-card 7456
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