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Theorem fin23lem7 8687
Description: Lemma for isfin2-2 8690. 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 3793 . . . 4  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
2 difss 3617 . . . . . . . 8  |-  ( A 
\  y )  C_  A
3 elpw2g 4600 . . . . . . . . 9  |-  ( A  e.  V  ->  (
( A  \  y
)  e.  ~P A  <->  ( A  \  y ) 
C_  A ) )
43ad2antrr 723 . . . . . . . 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 459 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  B  C_  ~P A )
76sselda 3489 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  y  e.  ~P A )
87elpwid 4009 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  y  C_  A )
9 dfss4 3729 . . . . . . . . 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 459 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  y  e.  B )
1210, 11eqeltrd 2542 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  ( A  \  ( A  \ 
y ) )  e.  B )
13 difeq2 3602 . . . . . . . . 9  |-  ( x  =  ( A  \ 
y )  ->  ( A  \  x )  =  ( A  \  ( A  \  y ) ) )
1413eleq1d 2523 . . . . . . . 8  |-  ( x  =  ( A  \ 
y )  ->  (
( A  \  x
)  e.  B  <->  ( A  \  ( A  \  y
) )  e.  B
) )
1514rspcev 3207 . . . . . . 7  |-  ( ( ( A  \  y
)  e.  ~P A  /\  ( A  \  ( A  \  y ) )  e.  B )  ->  E. x  e.  ~P  A ( A  \  x )  e.  B
)
165, 12, 15syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  E. x  e.  ~P  A ( A 
\  x )  e.  B )
1716ex 432 . . . . 5  |-  ( ( A  e.  V  /\  B  C_  ~P A )  ->  ( y  e.  B  ->  E. x  e.  ~P  A ( A 
\  x )  e.  B ) )
1817exlimdv 1729 . . . 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 1191 . 2  |-  ( ( A  e.  V  /\  B  C_  ~P A  /\  B  =/=  (/) )  ->  E. x  e.  ~P  A ( A 
\  x )  e.  B )
21 rabn0 3804 . 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 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   E.wrex 2805   {crab 2808    \ cdif 3458    C_ wss 3461   (/)c0 3783   ~Pcpw 3999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-pw 4001
This theorem is referenced by:  fin2i2  8689  isfin2-2  8690
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