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Theorem carddom 8172
Description: Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation i of [Quine] p. 232. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
carddom  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( card `  A
)  C_  ( card `  B )  <->  A  ~<_  B ) )

Proof of Theorem carddom
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 carddom2 7606 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
41, 2, 3syl2an 463 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( card `  A
)  C_  ( card `  B )  <->  A  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1685    C_ wss 3153   class class class wbr 4024    dom cdm 4688   ` cfv 5221    ~<_ cdom 6857   cardccrd 7564
This theorem is referenced by:  alephval2  8190  cartarlim  25316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-ac2 8085
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  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
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