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Theorem fin23lem7 8591
Description: Lemma for isfin2-2 8594. The componentwise complement of a nonempty collection of sets is nonempty. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
fin23lem7  |-  ( ( A  e.  V  /\  B  C_  ~P A  /\  B  =/=  (/) )  ->  { x  e.  ~P A  |  ( A  \  x )  e.  B }  =/=  (/) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem fin23lem7
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 n0 3749 . . . 4  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
2 difss 3586 . . . . . . . 8  |-  ( A 
\  y )  C_  A
3 elpw2g 4558 . . . . . . . . 9  |-  ( A  e.  V  ->  (
( A  \  y
)  e.  ~P A  <->  ( A  \  y ) 
C_  A ) )
43ad2antrr 725 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  (
( A  \  y
)  e.  ~P A  <->  ( A  \  y ) 
C_  A ) )
52, 4mpbiri 233 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  ( A  \  y )  e. 
~P A )
6 simpr 461 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  B  C_  ~P A )
76sselda 3459 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  y  e.  ~P A )
87elpwid 3973 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  y  C_  A )
9 dfss4 3687 . . . . . . . . 9  |-  ( y 
C_  A  <->  ( A  \  ( A  \  y
) )  =  y )
108, 9sylib 196 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  ( A  \  ( A  \ 
y ) )  =  y )
11 simpr 461 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  y  e.  B )
1210, 11eqeltrd 2540 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  ( A  \  ( A  \ 
y ) )  e.  B )
13 difeq2 3571 . . . . . . . . 9  |-  ( x  =  ( A  \ 
y )  ->  ( A  \  x )  =  ( A  \  ( A  \  y ) ) )
1413eleq1d 2521 . . . . . . . 8  |-  ( x  =  ( A  \ 
y )  ->  (
( A  \  x
)  e.  B  <->  ( A  \  ( A  \  y
) )  e.  B
) )
1514rspcev 3173 . . . . . . 7  |-  ( ( ( A  \  y
)  e.  ~P A  /\  ( A  \  ( A  \  y ) )  e.  B )  ->  E. x  e.  ~P  A ( A  \  x )  e.  B
)
165, 12, 15syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  E. x  e.  ~P  A ( A 
\  x )  e.  B )
1716ex 434 . . . . 5  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  ( y  e.  B  ->  E. x  e.  ~P  A ( A 
\  x )  e.  B ) )
1817exlimdv 1691 . . . 4  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  ( E. y 
y  e.  B  ->  E. x  e.  ~P  A ( A  \  x )  e.  B
) )
191, 18syl5bi 217 . . 3  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  ( B  =/=  (/)  ->  E. x  e.  ~P  A ( A  \  x )  e.  B
) )
20193impia 1185 . 2  |-  ( ( A  e.  V  /\  B  C_  ~P A  /\  B  =/=  (/) )  ->  E. x  e.  ~P  A ( A 
\  x )  e.  B )
21 rabn0 3760 . 2  |-  ( { x  e.  ~P A  |  ( A  \  x )  e.  B }  =/=  (/)  <->  E. x  e.  ~P  A ( A  \  x )  e.  B
)
2220, 21sylibr 212 1  |-  ( ( A  e.  V  /\  B  C_  ~P A  /\  B  =/=  (/) )  ->  { x  e.  ~P A  |  ( A  \  x )  e.  B }  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2645   E.wrex 2797   {crab 2800    \ cdif 3428    C_ wss 3431   (/)c0 3740   ~Pcpw 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-in 3438  df-ss 3445  df-nul 3741  df-pw 3965
This theorem is referenced by:  fin2i2  8593  isfin2-2  8594
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