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Theorem domtri2 8158
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 8146 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B ) )
2 cardon 8113 . . . 4  |-  ( card `  A )  e.  On
3 cardon 8113 . . . 4  |-  ( card `  B )  e.  On
4 ontri1 4752 . . . 4  |-  ( ( ( card `  A
)  e.  On  /\  ( card `  B )  e.  On )  ->  (
( card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
) )
52, 3, 4mp2an 672 . . 3  |-  ( (
card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
)
6 cardsdom2 8157 . . . . 5  |-  ( ( B  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  B )  e.  (
card `  A )  <->  B 
~<  A ) )
76ancoms 453 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  B )  e.  (
card `  A )  <->  B 
~<  A ) )
87notbid 294 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( -.  ( card `  B )  e.  ( card `  A
)  <->  -.  B  ~<  A ) )
95, 8syl5bb 257 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  B )  <->  -.  B  ~<  A ) )
101, 9bitr3d 255 1  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1756    C_ wss 3327   class class class wbr 4291   Oncon0 4718   dom cdm 4839   ` cfv 5417    ~<_ cdom 7307    ~< csdm 7308   cardccrd 8104
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 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
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 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-card 8108
This theorem is referenced by:  fidomtri  8162  harsdom  8164  infdif  8377  infdif2  8378  infunsdom1  8381  infunsdom  8382  infxp  8383  domtri  8719  canthp1lem2  8819  pwfseqlem4a  8827  pwfseqlem4  8828  gchaleph  8837  numinfctb  29457
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