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Theorem fin2i2 8687
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 753 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  A  e. FinII )
3 ssrab2 3578 . . . . . 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 755 . . . . . 6  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  B  =/=  (/) )
6 fin23lem7 8685 . . . . . 6  |-  ( ( A  e. FinII  /\  B  C_  ~P A  /\  B  =/=  (/) )  ->  { c  e.  ~P A  |  ( A  \  c )  e.  B }  =/=  (/) )
72, 1, 5, 6syl3anc 1223 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  { c  e.  ~P A  |  ( A  \  c )  e.  B }  =/=  (/) )
8 sorpsscmpl 6566 . . . . . 6  |-  ( [ C.]  Or  B  -> [ C.]  Or  { c  e.  ~P A  |  ( A  \ 
c )  e.  B } )
98ad2antll 728 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  -> [ C.]  Or  { c  e. 
~P A  |  ( A  \  c )  e.  B } )
10 fin2i 8664 . . . . 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 1224 . . . 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 6564 . . . . 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 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 3585 . . . 4  |-  ( z  =  ( A  \  m )  ->  (
w  C.  z  <->  w  C.  ( A  \  m ) ) )
16 psseq2 3585 . . . 4  |-  ( n  =  ( A  \  w )  ->  (
m  C.  n  <->  m  C.  ( A  \  w ) ) )
17 pssdifcom2 3906 . . . 4  |-  ( ( m  C_  A  /\  w  C_  A )  -> 
( w  C.  ( A  \  m )  <->  m  C.  ( A  \  w ) ) )
1815, 16, 17fin23lem11 8686 . . 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 60 . 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 6565 . . 3  |-  ( [ C.]  Or  B  ->  ( E. z  e.  B  A. w  e.  B  -.  w  C.  z  <->  |^| B  e.  B ) )
2120ad2antll 728 . 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 369    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808   {crab 2811    \ cdif 3466    C_ wss 3469    C. wpss 3470   (/)c0 3778   ~Pcpw 4003   U.cuni 4238   |^|cint 4275    Or wor 4792   [ C.] crpss 6554  FinIIcfin2 8648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-int 4276  df-br 4441  df-opab 4499  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-rpss 6555  df-fin2 8655
This theorem is referenced by:  isfin2-2  8688  fin23lem40  8720  fin2so  29603
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