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Theorem fin2i2 8730
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 754 . . 3  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B
) )  ->  B  C_ 
~P A )
2 simpll 752 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B
) )  ->  A  e. FinII
)
3 ssrab2 3524 . . . . . 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 756 . . . . . 6  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B
) )  ->  B  =/=  (/) )
6 fin23lem7 8728 . . . . . 6  |-  ( ( A  e. FinII  /\  B  C_  ~P A  /\  B  =/=  (/) )  ->  { c  e.  ~P A  |  ( A  \  c )  e.  B }  =/=  (/) )
72, 1, 5, 6syl3anc 1230 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B
) )  ->  { c  e.  ~P A  | 
( A  \  c
)  e.  B }  =/=  (/) )
8 sorpsscmpl 6573 . . . . . 6  |-  ( [ C.]  Or  B  -> [ C.]  Or  {
c  e.  ~P A  |  ( A  \ 
c )  e.  B } )
98ad2antll 727 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B
) )  -> [ C.]  Or  {
c  e.  ~P A  |  ( A  \ 
c )  e.  B } )
10 fin2i 8707 . . . . 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 1231 . . . 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 6571 . . . . 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 17 . . . 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 232 . . 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 3531 . . . 4  |-  ( z  =  ( A  \  m )  ->  (
w  C.  z  <->  w  C.  ( A  \  m ) ) )
16 psseq2 3531 . . . 4  |-  ( n  =  ( A  \  w )  ->  (
m  C.  n  <->  m  C.  ( A  \  w ) ) )
17 pssdifcom2 3858 . . . 4  |-  ( ( m  C_  A  /\  w  C_  A )  -> 
( w  C.  ( A  \  m )  <->  m  C.  ( A  \  w ) ) )
1815, 16, 17fin23lem11 8729 . . 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 59 . 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 6572 . . 3  |-  ( [ C.]  Or  B  ->  ( E. z  e.  B  A. w  e.  B  -.  w  C.  z  <->  |^| B  e.  B ) )
2120ad2antll 727 . 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 210 1  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B
) )  ->  |^| B  e.  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755   {crab 2758    \ cdif 3411    C_ wss 3414    C. wpss 3415   (/)c0 3738   ~Pcpw 3955   U.cuni 4191   |^|cint 4227    Or wor 4743   [ C.] crpss 6561  FinIIcfin2 8691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-int 4228  df-br 4396  df-opab 4454  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-rpss 6562  df-fin2 8698
This theorem is referenced by:  isfin2-2  8731  fin23lem40  8763  fin2so  31412
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