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

Theorem cardsdom 8173
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 8093 . 2  |-  ( A  e.  V  ->  A  e.  dom  card )
2 numth3 8093 . 2  |-  ( B  e.  W  ->  B  e.  dom  card )
3 cardsdom2 7617 . 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 1685   class class class wbr 4025   dom cdm 4689   ` cfv 5222    ~< csdm 6858   cardccrd 7564
This theorem is referenced by:  canth3  8179  inar1  8393  cardtar  25304  fnctartar3  25309
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-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-ac2 8085
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-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  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-iun 3909  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-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  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-isom 5231  df-iota 6253  df-riota 6300  df-recs 6384  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-card 7568  df-ac 7739
  Copyright terms: Public domain W3C validator