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Theorem carddom 8266
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 8187 . 2  |-  ( A  e.  V  ->  A  e.  dom  card )
2 numth3 8187 . 2  |-  ( B  e.  W  ->  B  e.  dom  card )
3 carddom2 7700 . 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 1710    C_ wss 3228   class class class wbr 4104   dom cdm 4771   ` cfv 5337    ~<_ cdom 6949   cardccrd 7658
This theorem is referenced by:  alephval2  8284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-ac2 8179
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-suc 4480  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-riota 6391  df-recs 6475  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-card 7662  df-ac 7833
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