MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  domtri2 Structured version   Unicode version

Theorem domtri2 7878
Description: Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
domtri2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )

Proof of Theorem domtri2
StepHypRef Expression
1 carddom2 7866 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
2 cardon 7833 . . . 4  |-  ( card `  A )  e.  On
3 cardon 7833 . . . 4  |-  ( card `  B )  e.  On
4 ontri1 4617 . . . 4  |-  ( ( ( card `  A
)  e.  On  /\  ( card `  B )  e.  On )  ->  (
( card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
) )
52, 3, 4mp2an 655 . . 3  |-  ( (
card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
)
6 cardsdom2 7877 . . . . 5  |-  ( ( B  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  B )  e.  (
card `  A )  <->  B 
~<  A ) )
76ancoms 441 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  B )  e.  (
card `  A )  <->  B 
~<  A ) )
87notbid 287 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( -.  ( card `  B )  e.  ( card `  A
)  <->  -.  B  ~<  A ) )
95, 8syl5bb 250 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  -.  B  ~<  A ) )
101, 9bitr3d 248 1  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726    C_ wss 3322   class class class wbr 4214   Oncon0 4583   dom cdm 4880   ` cfv 5456    ~<_ cdom 7109    ~< csdm 7110   cardccrd 7824
This theorem is referenced by:  fidomtri  7882  harsdom  7884  infdif  8091  infdif2  8092  infunsdom1  8095  infunsdom  8096  infxp  8097  domtri  8433  canthp1lem2  8530  pwfseqlem4a  8538  pwfseqlem4  8539  gchaleph  8552  numinfctb  27247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-card 7828
  Copyright terms: Public domain W3C validator