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Theorem fincssdom 8195
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 3425 . . . . . . 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 7285 . . . . . 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 7225 . . . . 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 7145 . . 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 1725    C_ wss 3312    C. wpss 3313   class class class wbr 4204    ~<_ cdom 7099    ~< csdm 7100   Fincfn 7101
This theorem is referenced by:  fin1a2lem11  8282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105
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