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Axiom ax-reg 7190
Description: Axiom of Regularity. An axiom of Zermelo-Fraenkel set theory. Also called the Axiom of Foundation. A rather non-intuitive axiom that denies more than it asserts, it states (in the form of zfreg 7193) that every non-empty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 7195). A stronger version that works for proper classes is proved as zfregs 7298. (Contributed by NM, 14-Aug-1993.)
Assertion
Ref Expression
ax-reg  |-  ( E. y  y  e.  x  ->  E. y ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) ) )
Distinct variable group:    x, y, z

Detailed syntax breakdown of Axiom ax-reg
StepHypRef Expression
1 vy . . . 4  set  y
2 vx . . . 4  set  x
31, 2wel 1622 . . 3  wff  y  e.  x
43, 1wex 1537 . 2  wff  E. y 
y  e.  x
5 vz . . . . . . 7  set  z
65, 1wel 1622 . . . . . 6  wff  z  e.  y
75, 2wel 1622 . . . . . . 7  wff  z  e.  x
87wn 5 . . . . . 6  wff  -.  z  e.  x
96, 8wi 6 . . . . 5  wff  ( z  e.  y  ->  -.  z  e.  x )
109, 5wal 1532 . . . 4  wff  A. z
( z  e.  y  ->  -.  z  e.  x )
113, 10wa 360 . . 3  wff  ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) )
1211, 1wex 1537 . 2  wff  E. y
( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x )
)
134, 12wi 6 1  wff  ( E. y  y  e.  x  ->  E. y ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) ) )
Colors of variables: wff set class
This axiom is referenced by:  axreg2  7191
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