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Theorem a9e 1892
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 1545 through ax-14 1689 and ax-17 1604, all axioms other than ax9 1890 are believed to be theorems of free logic, although the system without ax9 1890 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 1890 . 2  |-  -.  A. x  -.  x  =  y
2 df-ex 1530 . 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 1528   E.wex 1529
This theorem is referenced by:  equs4  1901  equvini  1929  dtrucor2  4208  axextnd  8208  ax13dfeq  23556  a9e2nd  27595  a9e2ndVD  27952  a9e2ndALT  27975  a12stdy1  28393
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1530
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