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Theorem carddom2 7626
Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8192, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
carddom2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )

Proof of Theorem carddom2
StepHypRef Expression
1 carddomi2 7619 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  ->  A  ~<_  B ) )
2 brdom2 6907 . . 3  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )
3 cardon 7593 . . . . . . . 8  |-  ( card `  A )  e.  On
43onelssi 4517 . . . . . . 7  |-  ( (
card `  B )  e.  ( card `  A
)  ->  ( card `  B )  C_  ( card `  A ) )
5 carddomi2 7619 . . . . . . . 8  |-  ( ( B  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  B )  C_  ( card `  A )  ->  B  ~<_  A ) )
65ancoms 439 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  B )  C_  ( card `  A )  ->  B  ~<_  A ) )
7 domnsym 7003 . . . . . . 7  |-  ( B  ~<_  A  ->  -.  A  ~<  B )
84, 6, 7syl56 30 . . . . . 6  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  B )  e.  (
card `  A )  ->  -.  A  ~<  B ) )
98con2d 107 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<  B  ->  -.  ( card `  B )  e.  (
card `  A )
) )
10 cardon 7593 . . . . . 6  |-  ( card `  B )  e.  On
11 ontri1 4442 . . . . . 6  |-  ( ( ( card `  A
)  e.  On  /\  ( card `  B )  e.  On )  ->  (
( card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
) )
123, 10, 11mp2an 653 . . . . 5  |-  ( (
card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
)
139, 12syl6ibr 218 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<  B  ->  ( card `  A
)  C_  ( card `  B ) ) )
14 carden2b 7616 . . . . . 6  |-  ( A 
~~  B  ->  ( card `  A )  =  ( card `  B
) )
15 eqimss 3243 . . . . . 6  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( card `  A )  C_  ( card `  B ) )
1614, 15syl 15 . . . . 5  |-  ( A 
~~  B  ->  ( card `  A )  C_  ( card `  B )
)
1716a1i 10 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~~  B  ->  ( card `  A
)  C_  ( card `  B ) ) )
1813, 17jaod 369 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( A 
~<  B  \/  A  ~~  B )  ->  ( card `  A )  C_  ( card `  B )
) )
192, 18syl5bi 208 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  ->  ( card `  A
)  C_  ( card `  B ) ) )
201, 19impbid 183 1  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039   Oncon0 4408   dom cdm 4705   ` cfv 5271    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   cardccrd 7584
This theorem is referenced by:  carduni  7630  carden2  7636  cardsdom2  7637  domtri2  7638  infxpidm2  7660  cardaleph  7732  infenaleph  7734  alephinit  7738  ficardun2  7845  ackbij2  7885  cfflb  7901  fin1a2lem9  8050  carddom  8192  pwfseqlem5  8301  hashdom  11377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-card 7588
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