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Theorem n0f 3793
Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3794 requires only that  x not be free in, rather than not occur in,  A. (Contributed by NM, 17-Oct-2003.)
Hypothesis
Ref Expression
n0f.1  |-  F/_ x A
Assertion
Ref Expression
n0f  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )

Proof of Theorem n0f
StepHypRef Expression
1 n0f.1 . . . . 5  |-  F/_ x A
2 nfcv 2629 . . . . 5  |-  F/_ x (/)
31, 2cleqf 2656 . . . 4  |-  ( A  =  (/)  <->  A. x ( x  e.  A  <->  x  e.  (/) ) )
4 noel 3789 . . . . . 6  |-  -.  x  e.  (/)
54nbn 347 . . . . 5  |-  ( -.  x  e.  A  <->  ( x  e.  A  <->  x  e.  (/) ) )
65albii 1620 . . . 4  |-  ( A. x  -.  x  e.  A  <->  A. x ( x  e.  A  <->  x  e.  (/) ) )
73, 6bitr4i 252 . . 3  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
87necon3abii 2727 . 2  |-  ( A  =/=  (/)  <->  -.  A. x  -.  x  e.  A
)
9 df-ex 1597 . 2  |-  ( E. x  x  e.  A  <->  -. 
A. x  -.  x  e.  A )
108, 9bitr4i 252 1  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   F/_wnfc 2615    =/= wne 2662   (/)c0 3785
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-nul 3786
This theorem is referenced by:  n0  3794  abn0  3804  cp  8308  ordtconlem1  27558
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