MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eq0 Unicode version

Theorem eq0 3602
Description: The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
Assertion
Ref Expression
eq0  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
Distinct variable group:    x, A

Proof of Theorem eq0
StepHypRef Expression
1 neq0 3598 . . 3  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
2 df-ex 1548 . . 3  |-  ( E. x  x  e.  A  <->  -. 
A. x  -.  x  e.  A )
31, 2bitri 241 . 2  |-  ( -.  A  =  (/)  <->  -.  A. x  -.  x  e.  A
)
43con4bii 289 1  |-  ( A  =  (/)  <->  A. 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   (/)c0 3588
This theorem is referenced by:  0el  3604  ssdif0  3646  difin0ss  3654  inssdif0  3655  ralf0  3694  disjiun  4162  0ex  4299  dm0  5042  reldm0  5046  uzwo  10495  uzwoOLD  10496  fzouzdisj  11124  hashgt0elex  11625  hausdiag  17630  rnelfmlem  17937  nninfnub  26345  prtlem14  26613  stoweidlem34  27650  stoweidlem44  27660  bnj1476  28924
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 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-dif 3283  df-nul 3589
  Copyright terms: Public domain W3C validator