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Axiom ax-pow 4160
Description: Axiom of Power Sets. An axiom of Zermelo-Fraenkel set theory. It states that a set  y exists that includes the power set of a given set  x i.e. contains every subset of  x. The variant axpow2 4162 uses explicit subset notation. A version using class notation is pwex 4165. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ax-pow  |-  E. y A. z ( A. w
( w  e.  z  ->  w  e.  x
)  ->  z  e.  y )
Distinct variable group:    x, y, z, w

Detailed syntax breakdown of Axiom ax-pow
StepHypRef Expression
1 vw . . . . . . 7  set  w
2 vz . . . . . . 7  set  z
31, 2wel 1622 . . . . . 6  wff  w  e.  z
4 vx . . . . . . 7  set  x
51, 4wel 1622 . . . . . 6  wff  w  e.  x
63, 5wi 6 . . . . 5  wff  ( w  e.  z  ->  w  e.  x )
76, 1wal 1532 . . . 4  wff  A. w
( w  e.  z  ->  w  e.  x
)
8 vy . . . . 5  set  y
92, 8wel 1622 . . . 4  wff  z  e.  y
107, 9wi 6 . . 3  wff  ( A. w ( w  e.  z  ->  w  e.  x )  ->  z  e.  y )
1110, 2wal 1532 . 2  wff  A. z
( A. w ( w  e.  z  ->  w  e.  x )  ->  z  e.  y )
1211, 8wex 1537 1  wff  E. y A. z ( A. w
( w  e.  z  ->  w  e.  x
)  ->  z  e.  y )
Colors of variables: wff set class
This axiom is referenced by:  zfpow  4161  axpow2  4162
  Copyright terms: Public domain W3C validator