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Theorem fin23lem7 8152
Description: Lemma for isfin2-2 8155. 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 3597 . . . 4  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
2 difss 3434 . . . . . . . 8  |-  ( A 
\  y )  C_  A
3 elpw2g 4323 . . . . . . . . 9  |-  ( A  e.  V  ->  (
( A  \  y
)  e.  ~P A  <->  ( A  \  y ) 
C_  A ) )
43ad2antrr 707 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  (
( A  \  y
)  e.  ~P A  <->  ( A  \  y ) 
C_  A ) )
52, 4mpbiri 225 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  ( A  \  y )  e. 
~P A )
6 simpr 448 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  B  C_  ~P A )
76sselda 3308 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  y  e.  ~P A )
87elpwid 3768 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  y  C_  A )
9 dfss4 3535 . . . . . . . . 9  |-  ( y 
C_  A  <->  ( A  \  ( A  \  y
) )  =  y )
108, 9sylib 189 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  ( A  \  ( A  \ 
y ) )  =  y )
11 simpr 448 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  y  e.  B )
1210, 11eqeltrd 2478 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  ( A  \  ( A  \ 
y ) )  e.  B )
13 difeq2 3419 . . . . . . . . 9  |-  ( x  =  ( A  \ 
y )  ->  ( A  \  x )  =  ( A  \  ( A  \  y ) ) )
1413eleq1d 2470 . . . . . . . 8  |-  ( x  =  ( A  \ 
y )  ->  (
( A  \  x
)  e.  B  <->  ( A  \  ( A  \  y
) )  e.  B
) )
1514rspcev 3012 . . . . . . 7  |-  ( ( ( A  \  y
)  e.  ~P A  /\  ( A  \  ( A  \  y ) )  e.  B )  ->  E. x  e.  ~P  A ( A  \  x )  e.  B
)
165, 12, 15syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  E. x  e.  ~P  A ( A 
\  x )  e.  B )
1716ex 424 . . . . 5  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  ( y  e.  B  ->  E. x  e.  ~P  A ( A 
\  x )  e.  B ) )
1817exlimdv 1643 . . . 4  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  ( E. y 
y  e.  B  ->  E. x  e.  ~P  A ( A  \  x )  e.  B
) )
191, 18syl5bi 209 . . 3  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  ( B  =/=  (/)  ->  E. x  e.  ~P  A ( A  \  x )  e.  B
) )
20193impia 1150 . 2  |-  ( ( A  e.  V  /\  B  C_  ~P A  /\  B  =/=  (/) )  ->  E. x  e.  ~P  A ( A 
\  x )  e.  B )
21 rabn0 3607 . 2  |-  ( { x  e.  ~P A  |  ( A  \  x )  e.  B }  =/=  (/)  <->  E. x  e.  ~P  A ( A  \  x )  e.  B
)
2220, 21sylibr 204 1  |-  ( ( A  e.  V  /\  B  C_  ~P A  /\  B  =/=  (/) )  ->  { x  e.  ~P A  |  ( A  \  x )  e.  B }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   {crab 2670    \ cdif 3277    C_ wss 3280   (/)c0 3588   ~Pcpw 3759
This theorem is referenced by:  fin2i2  8154  isfin2-2  8155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-in 3287  df-ss 3294  df-nul 3589  df-pw 3761
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