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Theorem mss 3989
Description: An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)
Assertion
Ref Expression
mss  |-  ( E. y  y  e.  A  ->  E. x ( x 
C_  A  /\  E. z  z  e.  x
) )
Distinct variable groups:    x, y    x, z    x, A, y
Allowed substitution hint:    A( z)

Proof of Theorem mss
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 vex 2605 . . . . 5  |-  y  e. 
_V
21snss 3524 . . . 4  |-  ( y  e.  A  <->  { y }  C_  A )
31snm 3518 . . . . 5  |-  E. w  w  e.  { y }
41snex 3965 . . . . . 6  |-  { y }  e.  _V
5 sseq1 3021 . . . . . . 7  |-  ( x  =  { y }  ->  ( x  C_  A 
<->  { y }  C_  A ) )
6 eleq2 2143 . . . . . . . 8  |-  ( x  =  { y }  ->  ( w  e.  x  <->  w  e.  { y } ) )
76exbidv 1747 . . . . . . 7  |-  ( x  =  { y }  ->  ( E. w  w  e.  x  <->  E. w  w  e.  { y } ) )
85, 7anbi12d 457 . . . . . 6  |-  ( x  =  { y }  ->  ( ( x 
C_  A  /\  E. w  w  e.  x
)  <->  ( { y }  C_  A  /\  E. w  w  e.  {
y } ) ) )
94, 8spcev 2693 . . . . 5  |-  ( ( { y }  C_  A  /\  E. w  w  e.  { y } )  ->  E. x
( x  C_  A  /\  E. w  w  e.  x ) )
103, 9mpan2 416 . . . 4  |-  ( { y }  C_  A  ->  E. x ( x 
C_  A  /\  E. w  w  e.  x
) )
112, 10sylbi 119 . . 3  |-  ( y  e.  A  ->  E. x
( x  C_  A  /\  E. w  w  e.  x ) )
1211exlimiv 1530 . 2  |-  ( E. y  y  e.  A  ->  E. x ( x 
C_  A  /\  E. w  w  e.  x
) )
13 elequ1 1641 . . . . 5  |-  ( z  =  w  ->  (
z  e.  x  <->  w  e.  x ) )
1413cbvexv 1837 . . . 4  |-  ( E. z  z  e.  x  <->  E. w  w  e.  x
)
1514anbi2i 445 . . 3  |-  ( ( x  C_  A  /\  E. z  z  e.  x
)  <->  ( x  C_  A  /\  E. w  w  e.  x ) )
1615exbii 1537 . 2  |-  ( E. x ( x  C_  A  /\  E. z  z  e.  x )  <->  E. x
( x  C_  A  /\  E. w  w  e.  x ) )
1712, 16sylibr 132 1  |-  ( E. y  y  e.  A  ->  E. x ( x 
C_  A  /\  E. z  z  e.  x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434    C_ wss 2974   {csn 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412
This theorem is referenced by: (None)
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