MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  carddom2 Structured version   Unicode version

Theorem carddom2 8257
Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8828, 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 8250 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  ->  A  ~<_  B ) )
2 brdom2 7448 . . 3  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )
3 cardon 8224 . . . . . . . 8  |-  ( card `  A )  e.  On
43onelssi 4934 . . . . . . 7  |-  ( (
card `  B )  e.  ( card `  A
)  ->  ( card `  B )  C_  ( card `  A ) )
5 carddomi2 8250 . . . . . . . 8  |-  ( ( B  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  B )  C_  ( card `  A )  ->  B  ~<_  A ) )
65ancoms 453 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  B )  C_  ( card `  A )  ->  B  ~<_  A ) )
7 domnsym 7546 . . . . . . 7  |-  ( B  ~<_  A  ->  -.  A  ~<  B )
84, 6, 7syl56 34 . . . . . 6  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  B )  e.  (
card `  A )  ->  -.  A  ~<  B ) )
98con2d 115 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<  B  ->  -.  ( card `  B )  e.  (
card `  A )
) )
10 cardon 8224 . . . . . 6  |-  ( card `  B )  e.  On
11 ontri1 4860 . . . . . 6  |-  ( ( ( card `  A
)  e.  On  /\  ( card `  B )  e.  On )  ->  (
( card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
) )
123, 10, 11mp2an 672 . . . . 5  |-  ( (
card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
)
139, 12syl6ibr 227 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<  B  ->  ( card `  A
)  C_  ( card `  B ) ) )
14 carden2b 8247 . . . . . 6  |-  ( A 
~~  B  ->  ( card `  A )  =  ( card `  B
) )
15 eqimss 3515 . . . . . 6  |-  ( (
card `  A )  =  ( card `  B
)  ->  ( card `  A )  C_  ( card `  B ) )
1614, 15syl 16 . . . . 5  |-  ( A 
~~  B  ->  ( card `  A )  C_  ( card `  B )
)
1716a1i 11 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~~  B  ->  ( card `  A
)  C_  ( card `  B ) ) )
1813, 17jaod 380 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( A 
~<  B  \/  A  ~~  B )  ->  ( card `  A )  C_  ( card `  B )
) )
192, 18syl5bi 217 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  ->  ( card `  A
)  C_  ( card `  B ) ) )
201, 19impbid 191 1  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3435   class class class wbr 4399   Oncon0 4826   dom cdm 4947   ` cfv 5525    ~~ cen 7416    ~<_ cdom 7417    ~< csdm 7418   cardccrd 8215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-card 8219
This theorem is referenced by:  carduni  8261  carden2  8267  cardsdom2  8268  domtri2  8269  infxpidm2  8293  cardaleph  8369  infenaleph  8371  alephinit  8375  ficardun2  8482  ackbij2  8522  cfflb  8538  fin1a2lem9  8687  carddom  8828  pwfseqlem5  8940  hashdom  12259
  Copyright terms: Public domain W3C validator