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Theorem n0f 3776
Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3777 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 2591 . . . . 5  |-  F/_ x (/)
31, 2cleqf 2618 . . . 4  |-  ( A  =  (/)  <->  A. x ( x  e.  A  <->  x  e.  (/) ) )
4 noel 3771 . . . . . 6  |-  -.  x  e.  (/)
54nbn 348 . . . . 5  |-  ( -.  x  e.  A  <->  ( x  e.  A  <->  x  e.  (/) ) )
65albii 1687 . . . 4  |-  ( A. x  -.  x  e.  A  <->  A. x ( x  e.  A  <->  x  e.  (/) ) )
73, 6bitr4i 255 . . 3  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
87necon3abii 2691 . 2  |-  ( A  =/=  (/)  <->  -.  A. x  -.  x  e.  A
)
9 df-ex 1660 . 2  |-  ( E. x  x  e.  A  <->  -. 
A. x  -.  x  e.  A )
108, 9bitr4i 255 1  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187   A.wal 1435    = wceq 1437   E.wex 1659    e. wcel 1870   F/_wnfc 2577    =/= wne 2625   (/)c0 3767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-v 3089  df-dif 3445  df-nul 3768
This theorem is referenced by:  n0  3777  abn0  3787  cp  8361  ordtconlem1  28569
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