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Theorem nnullss 4251
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 3477 . 2  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2 vex 2804 . . . . 5  |-  y  e. 
_V
32snss 3761 . . . 4  |-  ( y  e.  A  <->  { y }  C_  A )
42snnz 3757 . . . . 5  |-  { y }  =/=  (/)
5 snex 4232 . . . . . 6  |-  { y }  e.  _V
6 sseq1 3212 . . . . . . 7  |-  ( x  =  { y }  ->  ( x  C_  A 
<->  { y }  C_  A ) )
7 neeq1 2467 . . . . . . 7  |-  ( x  =  { y }  ->  ( x  =/=  (/) 
<->  { y }  =/=  (/) ) )
86, 7anbi12d 691 . . . . . 6  |-  ( x  =  { y }  ->  ( ( x 
C_  A  /\  x  =/=  (/) )  <->  ( {
y }  C_  A  /\  { y }  =/=  (/) ) ) )
95, 8spcev 2888 . . . . 5  |-  ( ( { y }  C_  A  /\  { y }  =/=  (/) )  ->  E. x
( x  C_  A  /\  x  =/=  (/) ) )
104, 9mpan2 652 . . . 4  |-  ( { y }  C_  A  ->  E. x ( x 
C_  A  /\  x  =/=  (/) ) )
113, 10sylbi 187 . . 3  |-  ( y  e.  A  ->  E. x
( x  C_  A  /\  x  =/=  (/) ) )
1211exlimiv 1624 . 2  |-  ( E. y  y  e.  A  ->  E. x ( x 
C_  A  /\  x  =/=  (/) ) )
131, 12sylbi 187 1  |-  ( A  =/=  (/)  ->  E. x
( x  C_  A  /\  x  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   {csn 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660
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