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Theorem a9e 1904
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 1547 through ax-14 1700 and ax-17 1606, all axioms other than ax9 1902 are believed to be theorems of free logic, although the system without ax9 1902 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 1902 . 2  |-  -.  A. x  -.  x  =  y
2 df-ex 1532 . 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 1530   E.wex 1531
This theorem is referenced by:  equs4  1912  equvini  1940  dtrucor2  4225  axextnd  8229  ax13dfeq  24226  a9e2nd  28623  a9e2ndVD  29000  a9e2ndALT  29023  a12stdy1  29748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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