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Theorem a9e 1568
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 1316 through ax-14 1386 and ax-17 1400, all axioms other than ax-9 1405 are believed to be theorems of free logic, although the system without ax-9 1405 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 1404 1 x x = y
Colors of variables: wff set class
Syntax hints:  wex 1362
This theorem was proved from axioms:  ax-i9 1404
This theorem is referenced by:  ax9o  1570  equid  1571  equs4  1595  equsal  1597  equsex  1598  equsexd  1599  spimt  1606  spimeh  1609  spimed  1610  equvini  1623  ax11v2  1683  ax11v  1690  ax11ev  1691  equs5or  1693  euequ1  1977
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