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Theorem nnullss 4699
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 3793 . 2  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2 vex 3109 . . . . 5  |-  y  e. 
_V
32snss 4140 . . . 4  |-  ( y  e.  A  <->  { y }  C_  A )
42snnz 4134 . . . . 5  |-  { y }  =/=  (/)
5 snex 4678 . . . . . 6  |-  { y }  e.  _V
6 sseq1 3510 . . . . . . 7  |-  ( x  =  { y }  ->  ( x  C_  A 
<->  { y }  C_  A ) )
7 neeq1 2735 . . . . . . 7  |-  ( x  =  { y }  ->  ( x  =/=  (/) 
<->  { y }  =/=  (/) ) )
86, 7anbi12d 708 . . . . . 6  |-  ( x  =  { y }  ->  ( ( x 
C_  A  /\  x  =/=  (/) )  <->  ( {
y }  C_  A  /\  { y }  =/=  (/) ) ) )
95, 8spcev 3198 . . . . 5  |-  ( ( { y }  C_  A  /\  { y }  =/=  (/) )  ->  E. x
( x  C_  A  /\  x  =/=  (/) ) )
104, 9mpan2 669 . . . 4  |-  ( { y }  C_  A  ->  E. x ( x 
C_  A  /\  x  =/=  (/) ) )
113, 10sylbi 195 . . 3  |-  ( y  e.  A  ->  E. x
( x  C_  A  /\  x  =/=  (/) ) )
1211exlimiv 1727 . 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 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649    C_ wss 3461   (/)c0 3783   {csn 4016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019
This theorem is referenced by: (None)
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