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Theorem a9e 1583
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 1333 through ax-14 1402 and ax-17 1416, all axioms other than ax-9 1421 are believed to be theorems of free logic, although the system without ax-9 1421 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
Ref Expression
a9e x x = y

Proof of Theorem a9e
StepHypRef Expression
1 ax-i9 1420 1 x x = y
Colors of variables: wff set class
Syntax hints:  wex 1378
This theorem was proved from axioms:  ax-i9 1420
This theorem is referenced by:  ax9o  1585  equid  1586  equs4  1610  equsal  1612  equsex  1613  equsexd  1614  spimt  1621  spimeh  1624  spimed  1625  equvini  1638  ax11v2  1698  ax11v  1705  ax11ev  1706  equs5or  1708  euequ1  1992
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