Theorem a9e 1951

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 1557 through ax-14 1714 and ax-17 1616, all axioms other than ax9 1949 are believed to be theorems of free logic, although the system without ax9 1949 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
a9e	⊢ ∃x x = y

Proof of Theorem a9e

Step	Hyp	Ref	Expression
1		ax9 1949	. 2 ⊢ ¬ ∀x ¬ x = y
2		df-ex 1542	. 2 ⊢ (∃x x = y ↔ ¬ ∀x ¬ x = y)
3	1, 2	mpbir 200	1 ⊢ ∃x x = y

Syntax hints:  ¬ wn 3  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  equs4  1959  equvini  1987  ax11vALT  2097  axi9  2330  dmi  4920