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Theorem fin2i2 8198
Description: A II-finite set contains minimal elements for every nonempty chain. (Contributed by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
fin2i2  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  |^| B  e.  B )

Proof of Theorem fin2i2
Dummy variables  c  m  n  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 732 . . 3  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  B  C_  ~P A )
2 simpll 731 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  A  e. FinII )
3 ssrab2 3428 . . . . . 6  |-  { c  e.  ~P A  | 
( A  \  c
)  e.  B }  C_ 
~P A
43a1i 11 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  { c  e.  ~P A  |  ( A  \  c )  e.  B }  C_  ~P A )
5 simprl 733 . . . . . 6  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  B  =/=  (/) )
6 fin23lem7 8196 . . . . . 6  |-  ( ( A  e. FinII  /\  B  C_  ~P A  /\  B  =/=  (/) )  ->  { c  e.  ~P A  |  ( A  \  c )  e.  B }  =/=  (/) )
72, 1, 5, 6syl3anc 1184 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  { c  e.  ~P A  |  ( A  \  c )  e.  B }  =/=  (/) )
8 sorpsscmpl 6533 . . . . . 6  |-  ( [ C.]  Or  B  -> [ C.]  Or  { c  e.  ~P A  |  ( A  \ 
c )  e.  B } )
98ad2antll 710 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  -> [ C.]  Or  { c  e. 
~P A  |  ( A  \  c )  e.  B } )
10 fin2i 8175 . . . . 5  |-  ( ( ( A  e. FinII  /\  {
c  e.  ~P A  |  ( A  \ 
c )  e.  B }  C_  ~P A )  /\  ( { c  e.  ~P A  | 
( A  \  c
)  e.  B }  =/=  (/)  /\ [ C.]  Or  {
c  e.  ~P A  |  ( A  \ 
c )  e.  B } ) )  ->  U. { c  e.  ~P A  |  ( A  \  c )  e.  B }  e.  { c  e.  ~P A  |  ( A  \  c )  e.  B } )
112, 4, 7, 9, 10syl22anc 1185 . . . 4  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  U. { c  e.  ~P A  |  ( A  \  c )  e.  B }  e.  { c  e.  ~P A  |  ( A  \  c )  e.  B } )
12 sorpssuni 6531 . . . . 5  |-  ( [ C.]  Or  { c  e.  ~P A  |  ( A  \  c )  e.  B }  ->  ( E. m  e.  { c  e.  ~P A  |  ( A  \  c )  e.  B } A. n  e.  {
c  e.  ~P A  |  ( A  \ 
c )  e.  B }  -.  m  C.  n  <->  U. { c  e.  ~P A  |  ( A  \  c )  e.  B }  e.  { c  e.  ~P A  |  ( A  \  c )  e.  B } ) )
139, 12syl 16 . . . 4  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  -> 
( E. m  e. 
{ c  e.  ~P A  |  ( A  \  c )  e.  B } A. n  e.  {
c  e.  ~P A  |  ( A  \ 
c )  e.  B }  -.  m  C.  n  <->  U. { c  e.  ~P A  |  ( A  \  c )  e.  B }  e.  { c  e.  ~P A  |  ( A  \  c )  e.  B } ) )
1411, 13mpbird 224 . . 3  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  E. m  e.  { c  e.  ~P A  | 
( A  \  c
)  e.  B } A. n  e.  { c  e.  ~P A  | 
( A  \  c
)  e.  B }  -.  m  C.  n )
15 psseq2 3435 . . . 4  |-  ( z  =  ( A  \  m )  ->  (
w  C.  z  <->  w  C.  ( A  \  m
) ) )
16 psseq2 3435 . . . 4  |-  ( n  =  ( A  \  w )  ->  (
m  C.  n  <->  m  C.  ( A  \  w
) ) )
17 pssdifcom2 3714 . . . 4  |-  ( ( m  C_  A  /\  w  C_  A )  -> 
( w  C.  ( A  \  m )  <->  m  C.  ( A  \  w
) ) )
1815, 16, 17fin23lem11 8197 . . 3  |-  ( B 
C_  ~P A  ->  ( E. m  e.  { c  e.  ~P A  | 
( A  \  c
)  e.  B } A. n  e.  { c  e.  ~P A  | 
( A  \  c
)  e.  B }  -.  m  C.  n  ->  E. z  e.  B  A. w  e.  B  -.  w  C.  z ) )
191, 14, 18sylc 58 . 2  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  E. z  e.  B  A. w  e.  B  -.  w  C.  z )
20 sorpssint 6532 . . 3  |-  ( [ C.]  Or  B  ->  ( E. z  e.  B  A. w  e.  B  -.  w  C.  z  <->  |^| B  e.  B ) )
2120ad2antll 710 . 2  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  -> 
( E. z  e.  B  A. w  e.  B  -.  w  C.  z 
<-> 
|^| B  e.  B
) )
2219, 21mpbid 202 1  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  |^| B  e.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709    \ cdif 3317    C_ wss 3320    C. wpss 3321   (/)c0 3628   ~Pcpw 3799   U.cuni 4015   |^|cint 4050    Or wor 4502   [ C.] crpss 6521  FinIIcfin2 8159
This theorem is referenced by:  isfin2-2  8199  fin23lem40  8231
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-rpss 6522  df-fin2 8166
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