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Theorem n0f 3579
Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3580 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 2523 . . . . 5  |-  F/_ x (/)
31, 2cleqf 2547 . . . 4  |-  ( A  =  (/)  <->  A. x ( x  e.  A  <->  x  e.  (/) ) )
4 noel 3575 . . . . . 6  |-  -.  x  e.  (/)
54nbn 337 . . . . 5  |-  ( -.  x  e.  A  <->  ( x  e.  A  <->  x  e.  (/) ) )
65albii 1572 . . . 4  |-  ( A. x  -.  x  e.  A  <->  A. x ( x  e.  A  <->  x  e.  (/) ) )
73, 6bitr4i 244 . . 3  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
87necon3abii 2580 . 2  |-  ( A  =/=  (/)  <->  -.  A. x  -.  x  e.  A
)
9 df-ex 1548 . 2  |-  ( E. x  x  e.  A  <->  -. 
A. x  -.  x  e.  A )
108, 9bitr4i 244 1  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717   F/_wnfc 2510    =/= wne 2550   (/)c0 3571
This theorem is referenced by:  n0  3580  abn0  3589  cp  7748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-v 2901  df-dif 3266  df-nul 3572
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