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Theorem a9e 1891
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 1544 through ax-14 1688 and ax-17 1603, all axioms other than ax9 1889 are believed to be theorems of free logic, although the system without ax9 1889 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 ax9 1889 . 2  |-  -.  A. x  -.  x  =  y
2 df-ex 1529 . 2  |-  ( E. x  x  =  y  <->  -.  A. x  -.  x  =  y )
31, 2mpbir 200 1  |-  E. x  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3   A.wal 1527   E.wex 1528
This theorem is referenced by:  equs4  1899  equvini  1927  dtrucor2  4209  axextnd  8213  ax13dfeq  24155  a9e2nd  28324  a9e2ndVD  28684  a9e2ndALT  28707  a12stdy1  29126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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