MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  n0f Structured version   Unicode version

Theorem n0f 3628
Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3629 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 2571 . . . . 5  |-  F/_ x (/)
31, 2cleqf 2595 . . . 4  |-  ( A  =  (/)  <->  A. x ( x  e.  A  <->  x  e.  (/) ) )
4 noel 3624 . . . . . 6  |-  -.  x  e.  (/)
54nbn 337 . . . . 5  |-  ( -.  x  e.  A  <->  ( x  e.  A  <->  x  e.  (/) ) )
65albii 1575 . . . 4  |-  ( A. x  -.  x  e.  A  <->  A. x ( x  e.  A  <->  x  e.  (/) ) )
73, 6bitr4i 244 . . 3  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
87necon3abii 2628 . 2  |-  ( A  =/=  (/)  <->  -.  A. x  -.  x  e.  A
)
9 df-ex 1551 . 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 1549   E.wex 1550    = wceq 1652    e. wcel 1725   F/_wnfc 2558    =/= wne 2598   (/)c0 3620
This theorem is referenced by:  n0  3629  abn0  3638  cp  7807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-nul 3621
  Copyright terms: Public domain W3C validator