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Theorem carddom 8363
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 8284 . 2  |-  ( A  e.  V  ->  A  e.  dom  card )
2 numth3 8284 . 2  |-  ( B  e.  W  ->  B  e.  dom  card )
3 carddom2 7798 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
41, 2, 3syl2an 464 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 177    /\ wa 359    e. wcel 1717    C_ wss 3264   class class class wbr 4154   dom cdm 4819   ` cfv 5395    ~<_ cdom 7044   cardccrd 7756
This theorem is referenced by:  alephval2  8381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-ac2 8277
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-suc 4529  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-riota 6486  df-recs 6570  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-card 7760  df-ac 7931
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