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Axiom ax-sep 4016
Description: The Axiom of Separation of IZF set theory. Axiom 6 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed, and with a  F/ y ph condition replaced by a distinct variable constraint between  y and  ph).

The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with  x  e.  z) so that it asserts the existence of a collection only if it is smaller than some other collection  z that already exists. This prevents Russell's paradox ru 2881. In some texts, this scheme is called "Aussonderung" or the Subset Axiom.

(Contributed by NM, 11-Sep-2006.)

Assertion
Ref Expression
ax-sep  |-  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
Distinct variable groups:    x, y, z    ph, y, z
Allowed substitution hint:    ph( x)

Detailed syntax breakdown of Axiom ax-sep
StepHypRef Expression
1 vx . . . . 5  setvar  x
2 vy . . . . 5  setvar  y
31, 2wel 1466 . . . 4  wff  x  e.  y
4 vz . . . . . 6  setvar  z
51, 4wel 1466 . . . . 5  wff  x  e.  z
6 wph . . . . 5  wff  ph
75, 6wa 103 . . . 4  wff  ( x  e.  z  /\  ph )
83, 7wb 104 . . 3  wff  ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
98, 1wal 1314 . 2  wff  A. x
( x  e.  y  <-> 
( x  e.  z  /\  ph ) )
109, 2wex 1453 1  wff  E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph )
)
Colors of variables: wff set class
This axiom is referenced by:  axsep2  4017  zfauscl  4018  bm1.3ii  4019  a9evsep  4020  axnul  4023  nalset  4028
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