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Theorem fincssdom 8771
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 1033 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  /\  -.  A  C_  B )  ->  A  e.  Fin )
2 simpr 468 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  /\  -.  A  C_  B )  ->  -.  A  C_  B )
3 simpl3 1035 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  /\  -.  A  C_  B )  ->  ( A  C_  B  \/  B  C_  A ) )
4 orel1 389 . . . . . . . 8  |-  ( -.  A  C_  B  ->  ( ( A  C_  B  \/  B  C_  A )  ->  B  C_  A
) )
52, 3, 4sylc 61 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  /\  -.  A  C_  B )  ->  B  C_  A )
6 dfpss3 3505 . . . . . . 7  |-  ( B 
C.  A  <->  ( B  C_  A  /\  -.  A  C_  B ) )
75, 2, 6sylanbrc 677 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  /\  -.  A  C_  B )  ->  B  C.  A )
8 php3 7776 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  C.  A )  ->  B  ~<  A )
91, 7, 8syl2anc 673 . . . . 5  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  /\  -.  A  C_  B )  ->  B  ~<  A )
109ex 441 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( -.  A  C_  B  ->  B  ~<  A ) )
11 domnsym 7716 . . . . 5  |-  ( A  ~<_  B  ->  -.  B  ~<  A )
1211con2i 124 . . . 4  |-  ( B 
~<  A  ->  -.  A  ~<_  B )
1310, 12syl6 33 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( -.  A  C_  B  ->  -.  A  ~<_  B ) )
1413con4d 108 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  ~<_  B  ->  A  C_  B ) )
15 ssdomg 7633 . . 3  |-  ( B  e.  Fin  ->  ( A  C_  B  ->  A  ~<_  B ) )
16153ad2ant2 1052 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  C_  B  ->  A  ~<_  B ) )
1714, 16impbid 195 1  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  C_  B  \/  B  C_  A ) )  -> 
( A  ~<_  B  <->  A  C_  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    e. wcel 1904    C_ wss 3390    C. wpss 3391   class class class wbr 4395    ~<_ cdom 7585    ~< csdm 7586   Fincfn 7587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  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-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591
This theorem is referenced by:  fin1a2lem11  8858
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