ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  axpow2 Unicode version

Theorem axpow2 3929
Description: A variant of the Axiom of Power Sets ax-pow 3927 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
Assertion
Ref Expression
axpow2  |-  E. y A. z ( z  C_  x  ->  z  e.  y )
Distinct variable group:    x, y, z

Proof of Theorem axpow2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax-pow 3927 . 2  |-  E. y A. z ( A. w
( w  e.  z  ->  w  e.  x
)  ->  z  e.  y )
2 dfss2 2934 . . . . 5  |-  ( z 
C_  x  <->  A. w
( w  e.  z  ->  w  e.  x
) )
32imbi1i 227 . . . 4  |-  ( ( z  C_  x  ->  z  e.  y )  <->  ( A. w ( w  e.  z  ->  w  e.  x )  ->  z  e.  y ) )
43albii 1359 . . 3  |-  ( A. z ( z  C_  x  ->  z  e.  y )  <->  A. z ( A. w ( w  e.  z  ->  w  e.  x )  ->  z  e.  y ) )
54exbii 1496 . 2  |-  ( E. y A. z ( z  C_  x  ->  z  e.  y )  <->  E. y A. z ( A. w
( w  e.  z  ->  w  e.  x
)  ->  z  e.  y ) )
61, 5mpbir 134 1  |-  E. y A. z ( z  C_  x  ->  z  e.  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1241   E.wex 1381    C_ wss 2917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-pow 3927
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931
This theorem is referenced by:  axpow3  3930  pwex  3932
  Copyright terms: Public domain W3C validator