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Theorem fincssdom 8137
Description: In a chain of finite sets, dominance and subset coincide. (Contributed by Stefan O'Rear, 8-Nov-2014.)
Assertion
Ref Expression
fincssdom  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  ~<_  B  <->  A  C_  B
) )

Proof of Theorem fincssdom
StepHypRef Expression
1 simpl1 960 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  /\  -.  A  C_  B )  ->  A  e.  Fin )
2 simpr 448 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  /\  -.  A  C_  B )  ->  -.  A  C_  B )
3 simpl3 962 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  /\  -.  A  C_  B )  ->  ( A  C_  B  \/  B  C_  A ) )
4 orel1 372 . . . . . . . 8  |-  ( -.  A  C_  B  ->  ( ( A  C_  B  \/  B  C_  A )  ->  B  C_  A
) )
52, 3, 4sylc 58 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  /\  -.  A  C_  B )  ->  B  C_  A )
6 dfpss3 3377 . . . . . . 7  |-  ( B 
C.  A  <->  ( B  C_  A  /\  -.  A  C_  B ) )
75, 2, 6sylanbrc 646 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  /\  -.  A  C_  B )  ->  B  C.  A )
8 php3 7230 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  C.  A )  ->  B  ~<  A )
91, 7, 8syl2anc 643 . . . . 5  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  /\  -.  A  C_  B )  ->  B  ~<  A )
109ex 424 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( -.  A  C_  B  ->  B  ~<  A ) )
11 domnsym 7170 . . . . 5  |-  ( A  ~<_  B  ->  -.  B  ~<  A )
1211con2i 114 . . . 4  |-  ( B 
~<  A  ->  -.  A  ~<_  B )
1310, 12syl6 31 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( -.  A  C_  B  ->  -.  A  ~<_  B ) )
1413con4d 99 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  ~<_  B  ->  A  C_  B ) )
15 ssdomg 7090 . . 3  |-  ( B  e.  Fin  ->  ( A  C_  B  ->  A  ~<_  B ) )
16153ad2ant2 979 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  C_  B  ->  A  ~<_  B ) )
1714, 16impbid 184 1  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  ~<_  B  <->  A  C_  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    e. wcel 1717    C_ wss 3264    C. wpss 3265   class class class wbr 4154    ~<_ cdom 7044    ~< csdm 7045   Fincfn 7046
This theorem is referenced by:  fin1a2lem11  8224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050
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