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Theorem axreg 6770
Description: Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
axreg  |-  ( x  e.  y  ->  E. x
( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y
) ) )
Distinct variable group:    x, y, z

Proof of Theorem axreg
StepHypRef Expression
1 ax-reg 6769 . 2  |-  ( E. x  x  e.  y  ->  E. x ( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y ) ) )
2119.23bi 1662 1  |-  ( x  e.  y  ->  E. x
( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 357   A.wal 1450   E.wex 1455    e. wcel 1538
This theorem is referenced by:  zfregcl  6771  axregndlem2  7677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-4 1606  ax-reg 6769
This theorem depends on definitions:  df-bi 175  df-ex 1456
Copyright terms: Public domain