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Theorem a9e 1817
Description: At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1533 through ax-14 1626 and ax-17 1628, all axioms other than ax-9 1684 are believed to be theorems of free logic, although the system without ax-9 1684 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
a9e  |-  E. x  x  =  y

Proof of Theorem a9e
StepHypRef Expression
1 ax-9 1684 . 2  |-  -.  A. x  -.  x  =  y
2 df-ex 1538 . 2  |-  ( E. x  x  =  y  <->  -.  A. x  -.  x  =  y )
31, 2mpbir 202 1  |-  E. x  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 5   A.wal 1532   E.wex 1537
This theorem is referenced by:  equid1  1820  equs4  1849  equvini  1879  ax11v2  1935  ax11v2-o  1936  pm11.07  2075  ax11eq  2105  ax11el  2106  ax11inda  2113  euequ1  2201  dtrucor2  4103  snnex  4415  relop  4741  dmi  4800  1st2val  5997  2nd2val  5998  axextnd  8093  ax13dfeq  23323  ax10-16  26773  a9e2eq  27113  a9e2nd  27114  relopabVD  27464  a9e2eqVD  27470  a9e2ndVD  27471  a9e2ndALT  27494  bnj1468  27664  bnj1014  27778  a12stdy1  28030
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-9 1684
This theorem depends on definitions:  df-bi 179  df-ex 1538
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