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Theorem fin2i2 7960
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 731 . . 3  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  B  C_  ~P A )
2 simpll 730 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  A  e. FinII )
3 ssrab2 3271 . . . . . 6  |-  { c  e.  ~P A  | 
( A  \  c
)  e.  B }  C_ 
~P A
43a1i 10 . . . . 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 732 . . . . . 6  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  B  =/=  (/) )
6 fin23lem7 7958 . . . . . 6  |-  ( ( A  e. FinII  /\  B  C_  ~P A  /\  B  =/=  (/) )  ->  { c  e.  ~P A  |  ( A  \  c )  e.  B }  =/=  (/) )
72, 1, 5, 6syl3anc 1182 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  { c  e.  ~P A  |  ( A  \  c )  e.  B }  =/=  (/) )
8 sorpsscmpl 6304 . . . . . 6  |-  ( [ C.]  Or  B  -> [ C.]  Or  { c  e.  ~P A  |  ( A  \ 
c )  e.  B } )
98ad2antll 709 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  -> [ C.]  Or  { c  e. 
~P A  |  ( A  \  c )  e.  B } )
10 fin2i 7937 . . . . 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 1183 . . . 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 6302 . . . . 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 15 . . . 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 223 . . 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 3277 . . . 4  |-  ( z  =  ( A  \  m )  ->  (
w  C.  z  <->  w  C.  ( A  \  m
) ) )
16 psseq2 3277 . . . 4  |-  ( n  =  ( A  \  w )  ->  (
m  C.  n  <->  m  C.  ( A  \  w
) ) )
17 pssdifcom2 3553 . . . 4  |-  ( ( m  C_  A  /\  w  C_  A )  -> 
( w  C.  ( A  \  m )  <->  m  C.  ( A  \  w
) ) )
1815, 16, 17fin23lem11 7959 . . 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 56 . 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 6303 . . 3  |-  ( [ C.]  Or  B  ->  ( E. z  e.  B  A. w  e.  B  -.  w  C.  z  <->  |^| B  e.  B ) )
2120ad2antll 709 . 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 201 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 176    /\ wa 358    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    \ cdif 3162    C_ wss 3165    C. wpss 3166   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   |^|cint 3878    Or wor 4329   [ C.] crpss 6292  FinIIcfin2 7921
This theorem is referenced by:  isfin2-2  7961  fin23lem40  7993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-rpss 6293  df-fin2 7928
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