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Theorem fin2i2 7944
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 3258 . . . . . 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 7942 . . . . . 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 6288 . . . . . 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 7921 . . . . 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 6286 . . . . 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 3264 . . . 4  |-  ( z  =  ( A  \  m )  ->  (
w  C.  z  <->  w  C.  ( A  \  m
) ) )
16 psseq2 3264 . . . 4  |-  ( n  =  ( A  \  w )  ->  (
m  C.  n  <->  m  C.  ( A  \  w
) ) )
17 pssdifcom2 3540 . . . 4  |-  ( ( m  C_  A  /\  w  C_  A )  -> 
( w  C.  ( A  \  m )  <->  m  C.  ( A  \  w
) ) )
1815, 16, 17fin23lem11 7943 . . 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 6287 . . 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 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    \ cdif 3149    C_ wss 3152    C. wpss 3153   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   |^|cint 3862    Or wor 4313   [ C.] crpss 6276  FinIIcfin2 7905
This theorem is referenced by:  isfin2-2  7945  fin23lem40  7977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-rpss 6277  df-fin2 7912
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