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

Theorem n0f 3600
Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3601 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 2544 . . . . 5  |-  F/_ x (/)
31, 2cleqf 2568 . . . 4  |-  ( A  =  (/)  <->  A. x ( x  e.  A  <->  x  e.  (/) ) )
4 noel 3596 . . . . . 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 2601 . 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 1721   F/_wnfc 2531    =/= wne 2571   (/)c0 3592
This theorem is referenced by:  n0  3601  abn0  3610  cp  7775
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-v 2922  df-dif 3287  df-nul 3593
  Copyright terms: Public domain W3C validator