MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardsdomel Unicode version

Theorem cardsdomel 7540
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 7519 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
2 ensym 6843 . . . . . . 7  |-  ( (
card `  B )  ~~  B  ->  B  ~~  ( card `  B )
)
31, 2syl 17 . . . . . 6  |-  ( B  e.  dom  card  ->  B 
~~  ( card `  B
) )
4 sdomentr 6928 . . . . . 6  |-  ( ( A  ~<  B  /\  B  ~~  ( card `  B
) )  ->  A  ~<  ( card `  B
) )
53, 4sylan2 462 . . . . 5  |-  ( ( A  ~<  B  /\  B  e.  dom  card )  ->  A  ~<  ( card `  B ) )
6 ssdomg 6840 . . . . . . . 8  |-  ( A  e.  On  ->  (
( card `  B )  C_  A  ->  ( card `  B )  ~<_  A ) )
7 cardon 7510 . . . . . . . . 9  |-  ( card `  B )  e.  On
8 domtriord 6940 . . . . . . . . 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 4363 . . . . . . . 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 7539 . 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 3094   class class class wbr 3963   Oncon0 4329   dom cdm 4626   ` cfv 4638    ~~ cen 6793    ~<_ cdom 6794    ~< csdm 6795   cardccrd 7501
This theorem is referenced by:  iscard  7541  cardval2  7557  infxpenlem  7574  alephnbtwn  7631  alephnbtwn2  7632  alephord2  7636  alephsdom  7646  pwsdompw  7763  inaprc  8391
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 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-card 7505
  Copyright terms: Public domain W3C validator