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Theorem fidomtri 7642
Description: Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
fidomtri  |-  ( ( A  e.  Fin  /\  B  e.  V )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )

Proof of Theorem fidomtri
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 domnsym 7003 . 2  |-  ( A  ~<_  B  ->  -.  B  ~<  A )
2 finnum 7597 . . . . . 6  |-  ( A  e.  Fin  ->  A  e.  dom  card )
32adantr 451 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  V )  ->  A  e.  dom  card )
4 finnum 7597 . . . . 5  |-  ( B  e.  Fin  ->  B  e.  dom  card )
5 domtri2 7638 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
63, 4, 5syl2an 463 . . . 4  |-  ( ( ( A  e.  Fin  /\  B  e.  V )  /\  B  e.  Fin )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
76biimprd 214 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  V )  /\  B  e.  Fin )  ->  ( -.  B  ~<  A  ->  A  ~<_  B ) )
8 isinffi 7641 . . . . . . 7  |-  ( ( -.  B  e.  Fin  /\  A  e.  Fin )  ->  E. a  a : A -1-1-> B )
98ancoms 439 . . . . . 6  |-  ( ( A  e.  Fin  /\  -.  B  e.  Fin )  ->  E. a  a : A -1-1-> B )
109adantlr 695 . . . . 5  |-  ( ( ( A  e.  Fin  /\  B  e.  V )  /\  -.  B  e. 
Fin )  ->  E. a 
a : A -1-1-> B
)
11 brdomg 6888 . . . . . 6  |-  ( B  e.  V  ->  ( A  ~<_  B  <->  E. a 
a : A -1-1-> B
) )
1211ad2antlr 707 . . . . 5  |-  ( ( ( A  e.  Fin  /\  B  e.  V )  /\  -.  B  e. 
Fin )  ->  ( A  ~<_  B  <->  E. a 
a : A -1-1-> B
) )
1310, 12mpbird 223 . . . 4  |-  ( ( ( A  e.  Fin  /\  B  e.  V )  /\  -.  B  e. 
Fin )  ->  A  ~<_  B )
1413a1d 22 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  V )  /\  -.  B  e. 
Fin )  ->  ( -.  B  ~<  A  ->  A  ~<_  B ) )
157, 14pm2.61dan 766 . 2  |-  ( ( A  e.  Fin  /\  B  e.  V )  ->  ( -.  B  ~<  A  ->  A  ~<_  B ) )
161, 15impbid2 195 1  |-  ( ( A  e.  Fin  /\  B  e.  V )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    e. wcel 1696   class class class wbr 4039   dom cdm 4705   -1-1->wf1 5268    ~<_ cdom 6877    ~< csdm 6878   Fincfn 6879   cardccrd 7584
This theorem is referenced by:  fidomtri2  7643  fin56  8035  hauspwdom  17243  harinf  27230
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-lim 4413  df-suc 4414  df-om 4673  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-fin 6883  df-card 7588
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