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

Theorem carddom2 8139
Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8710, 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 8132 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  ->  A  ~<_  B ) )
2 brdom2 7331 . . 3  |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B
) )
3 cardon 8106 . . . . . . . 8  |-  ( card `  A )  e.  On
43onelssi 4822 . . . . . . 7  |-  ( (
card `  B )  e.  ( card `  A
)  ->  ( card `  B )  C_  ( card `  A ) )
5 carddomi2 8132 . . . . . . . 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 7429 . . . . . . 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 8106 . . . . . 6  |-  ( card `  B )  e.  On
11 ontri1 4748 . . . . . 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 8129 . . . . . 6  |-  ( A 
~~  B  ->  ( card `  A )  =  ( card `  B
) )
15 eqimss 3403 . . . . . 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 1369    e. wcel 1756    C_ wss 3323   class class class wbr 4287   Oncon0 4714   dom cdm 4835   ` cfv 5413    ~~ cen 7299    ~<_ cdom 7300    ~< csdm 7301   cardccrd 8097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-card 8101
This theorem is referenced by:  carduni  8143  carden2  8149  cardsdom2  8150  domtri2  8151  infxpidm2  8175  cardaleph  8251  infenaleph  8253  alephinit  8257  ficardun2  8364  ackbij2  8404  cfflb  8420  fin1a2lem9  8569  carddom  8710  pwfseqlem5  8822  hashdom  12134
  Copyright terms: Public domain W3C validator