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Theorem nnullss 4654
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 3746 . 2  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2 vex 3073 . . . . 5  |-  y  e. 
_V
32snss 4099 . . . 4  |-  ( y  e.  A  <->  { y }  C_  A )
42snnz 4093 . . . . 5  |-  { y }  =/=  (/)
5 snex 4633 . . . . . 6  |-  { y }  e.  _V
6 sseq1 3477 . . . . . . 7  |-  ( x  =  { y }  ->  ( x  C_  A 
<->  { y }  C_  A ) )
7 neeq1 2729 . . . . . . 7  |-  ( x  =  { y }  ->  ( x  =/=  (/) 
<->  { y }  =/=  (/) ) )
86, 7anbi12d 710 . . . . . 6  |-  ( x  =  { y }  ->  ( ( x 
C_  A  /\  x  =/=  (/) )  <->  ( {
y }  C_  A  /\  { y }  =/=  (/) ) ) )
95, 8spcev 3162 . . . . 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 1689 . 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 1370   E.wex 1587    e. wcel 1758    =/= wne 2644    C_ wss 3428   (/)c0 3737   {csn 3977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-sn 3978  df-pr 3980
This theorem is referenced by: (None)
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