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Theorem nnullss 4709
Description: A nonempty class (even if proper) has a nonempty 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 3794 . 2  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2 vex 3116 . . . . 5  |-  y  e. 
_V
32snss 4151 . . . 4  |-  ( y  e.  A  <->  { y }  C_  A )
42snnz 4145 . . . . 5  |-  { y }  =/=  (/)
5 snex 4688 . . . . . 6  |-  { y }  e.  _V
6 sseq1 3525 . . . . . . 7  |-  ( x  =  { y }  ->  ( x  C_  A 
<->  { y }  C_  A ) )
7 neeq1 2748 . . . . . . 7  |-  ( x  =  { y }  ->  ( x  =/=  (/) 
<->  { y }  =/=  (/) ) )
86, 7anbi12d 710 . . . . . 6  |-  ( x  =  { y }  ->  ( ( x 
C_  A  /\  x  =/=  (/) )  <->  ( {
y }  C_  A  /\  { y }  =/=  (/) ) ) )
95, 8spcev 3205 . . . . 5  |-  ( ( { y }  C_  A  /\  { y }  =/=  (/) )  ->  E. x
( x  C_  A  /\  x  =/=  (/) ) )
104, 9mpan2 671 . . . 4  |-  ( { y }  C_  A  ->  E. x ( x 
C_  A  /\  x  =/=  (/) ) )
113, 10sylbi 195 . . 3  |-  ( y  e.  A  ->  E. x
( x  C_  A  /\  x  =/=  (/) ) )
1211exlimiv 1698 . 2  |-  ( E. y  y  e.  A  ->  E. x ( x 
C_  A  /\  x  =/=  (/) ) )
131, 12sylbi 195 1  |-  ( A  =/=  (/)  ->  E. x
( x  C_  A  /\  x  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662    C_ wss 3476   (/)c0 3785   {csn 4027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-pr 4030
This theorem is referenced by: (None)
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