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Theorem cardsdom 8131
Description: Two sets have the strict dominance relationship iff their cardinalities have the membership relationship. Corollary 19.7(2) of [Eisenberg] p. 310. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
cardsdom  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( card `  A
)  e.  ( card `  B )  <->  A  ~<  B ) )

Proof of Theorem cardsdom
StepHypRef Expression
1 numth3 8051 . 2  |-  ( A  e.  V  ->  A  e.  dom  card )
2 numth3 8051 . 2  |-  ( B  e.  W  ->  B  e.  dom  card )
3 cardsdom2 7575 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  e.  (
card `  B )  <->  A 
~<  B ) )
41, 2, 3syl2an 465 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( card `  A
)  e.  ( card `  B )  <->  A  ~<  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1621   class class class wbr 3983   dom cdm 4647   ` cfv 4659    ~< csdm 6816   cardccrd 7522
This theorem is referenced by:  canth3  8137  inar1  8351  cardtar  25257  fnctartar3  25262
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-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-ac2 8043
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 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-suc 4356  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-iota 6211  df-riota 6258  df-recs 6342  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-card 7526  df-ac 7697
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