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Theorem fin23lem7 7942
Description: Lemma for isfin2-2 7945. 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 3464 . . . 4  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
2 difss 3303 . . . . . . . 8  |-  ( A 
\  y )  C_  A
3 elpw2g 4174 . . . . . . . . 9  |-  ( A  e.  V  ->  (
( A  \  y
)  e.  ~P A  <->  ( A  \  y ) 
C_  A ) )
43ad2antrr 706 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  (
( A  \  y
)  e.  ~P A  <->  ( A  \  y ) 
C_  A ) )
52, 4mpbiri 224 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  ( A  \  y )  e. 
~P A )
6 simpr 447 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  B  C_  ~P A )
76sselda 3180 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  y  e.  ~P A )
8 elpwi 3633 . . . . . . . . . 10  |-  ( y  e.  ~P A  -> 
y  C_  A )
97, 8syl 15 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  y  C_  A )
10 dfss4 3403 . . . . . . . . 9  |-  ( y 
C_  A  <->  ( A  \  ( A  \  y
) )  =  y )
119, 10sylib 188 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  ( A  \  ( A  \ 
y ) )  =  y )
12 simpr 447 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  y  e.  B )
1311, 12eqeltrd 2357 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  ( A  \  ( A  \ 
y ) )  e.  B )
14 difeq2 3288 . . . . . . . . 9  |-  ( x  =  ( A  \ 
y )  ->  ( A  \  x )  =  ( A  \  ( A  \  y ) ) )
1514eleq1d 2349 . . . . . . . 8  |-  ( x  =  ( A  \ 
y )  ->  (
( A  \  x
)  e.  B  <->  ( A  \  ( A  \  y
) )  e.  B
) )
1615rspcev 2884 . . . . . . 7  |-  ( ( ( A  \  y
)  e.  ~P A  /\  ( A  \  ( A  \  y ) )  e.  B )  ->  E. x  e.  ~P  A ( A  \  x )  e.  B
)
175, 13, 16syl2anc 642 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  E. x  e.  ~P  A ( A 
\  x )  e.  B )
1817ex 423 . . . . 5  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  ( y  e.  B  ->  E. x  e.  ~P  A ( A 
\  x )  e.  B ) )
1918exlimdv 1664 . . . 4  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  ( E. y 
y  e.  B  ->  E. x  e.  ~P  A ( A  \  x )  e.  B
) )
201, 19syl5bi 208 . . 3  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  ( B  =/=  (/)  ->  E. x  e.  ~P  A ( A  \  x )  e.  B
) )
21203impia 1148 . 2  |-  ( ( A  e.  V  /\  B  C_  ~P A  /\  B  =/=  (/) )  ->  E. x  e.  ~P  A ( A 
\  x )  e.  B )
22 rabn0 3474 . 2  |-  ( { x  e.  ~P A  |  ( A  \  x )  e.  B }  =/=  (/)  <->  E. x  e.  ~P  A ( A  \  x )  e.  B
)
2321, 22sylibr 203 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 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547    \ cdif 3149    C_ wss 3152   (/)c0 3455   ~Pcpw 3625
This theorem is referenced by:  fin2i2  7944  isfin2-2  7945
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  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-in 3159  df-ss 3166  df-nul 3456  df-pw 3627
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