MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axpowndlem1 Unicode version

Theorem axpowndlem1 8235
Description: Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
Assertion
Ref Expression
axpowndlem1  |-  ( A. x  x  =  y  ->  ( -.  x  =  y  ->  E. x A. y ( A. x
( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x ) ) )

Proof of Theorem axpowndlem1
StepHypRef Expression
1 pm2.24 101 . 2  |-  ( x  =  y  ->  ( -.  x  =  y  ->  E. x A. y
( A. x ( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x ) ) )
21sps 1751 1  |-  ( A. x  x  =  y  ->  ( -.  x  =  y  ->  E. x A. y ( A. x
( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696
This theorem is referenced by:  axpowndlem3  8237  axpownd  8239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
  Copyright terms: Public domain W3C validator