MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem7 Unicode version

Theorem fin23lem7 8032
Description: Lemma for isfin2-2 8035. 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 3540 . . . 4  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
2 difss 3379 . . . . . . . 8  |-  ( A 
\  y )  C_  A
3 elpw2g 4255 . . . . . . . . 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 3256 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  y  e.  ~P A )
8 elpwi 3709 . . . . . . . . . 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 3479 . . . . . . . . 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 2432 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  ~P A
)  /\  y  e.  B )  ->  ( A  \  ( A  \ 
y ) )  e.  B )
14 difeq2 3364 . . . . . . . . 9  |-  ( x  =  ( A  \ 
y )  ->  ( A  \  x )  =  ( A  \  ( A  \  y ) ) )
1514eleq1d 2424 . . . . . . . 8  |-  ( x  =  ( A  \ 
y )  ->  (
( A  \  x
)  e.  B  <->  ( A  \  ( A  \  y
) )  e.  B
) )
1615rspcev 2960 . . . . . . 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 1636 . . . 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 3550 . 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 1541    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620   {crab 2623    \ cdif 3225    C_ wss 3228   (/)c0 3531   ~Pcpw 3701
This theorem is referenced by:  fin2i2  8034  isfin2-2  8035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-in 3235  df-ss 3242  df-nul 3532  df-pw 3703
  Copyright terms: Public domain W3C validator