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Theorem nnullss 4366
Description: A non-empty class (even if proper) has a non-empty subset. (Contributed by NM, 23-Aug-2003.)
Assertion
Ref Expression
nnullss  |-  ( A  =/=  (/)  ->  E. x
( x  C_  A  /\  x  =/=  (/) ) )
Distinct variable group:    x, A

Proof of Theorem nnullss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 n0 3580 . 2  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2 vex 2902 . . . . 5  |-  y  e. 
_V
32snss 3869 . . . 4  |-  ( y  e.  A  <->  { y }  C_  A )
42snnz 3865 . . . . 5  |-  { y }  =/=  (/)
5 snex 4346 . . . . . 6  |-  { y }  e.  _V
6 sseq1 3312 . . . . . . 7  |-  ( x  =  { y }  ->  ( x  C_  A 
<->  { y }  C_  A ) )
7 neeq1 2558 . . . . . . 7  |-  ( x  =  { y }  ->  ( x  =/=  (/) 
<->  { y }  =/=  (/) ) )
86, 7anbi12d 692 . . . . . 6  |-  ( x  =  { y }  ->  ( ( x 
C_  A  /\  x  =/=  (/) )  <->  ( {
y }  C_  A  /\  { y }  =/=  (/) ) ) )
95, 8spcev 2986 . . . . 5  |-  ( ( { y }  C_  A  /\  { y }  =/=  (/) )  ->  E. x
( x  C_  A  /\  x  =/=  (/) ) )
104, 9mpan2 653 . . . 4  |-  ( { y }  C_  A  ->  E. x ( x 
C_  A  /\  x  =/=  (/) ) )
113, 10sylbi 188 . . 3  |-  ( y  e.  A  ->  E. x
( x  C_  A  /\  x  =/=  (/) ) )
1211exlimiv 1641 . 2  |-  ( E. y  y  e.  A  ->  E. x ( x 
C_  A  /\  x  =/=  (/) ) )
131, 12sylbi 188 1  |-  ( A  =/=  (/)  ->  E. x
( x  C_  A  /\  x  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2550    C_ wss 3263   (/)c0 3571   {csn 3757
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-sn 3763  df-pr 3764
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