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

Theorem pwssb 4120
Description: Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
Assertion
Ref Expression
pwssb  |-  ( A 
C_  ~P B  <->  A. x  e.  A  x  C_  B
)
Distinct variable groups:    x, A    x, B

Proof of Theorem pwssb
StepHypRef Expression
1 sspwuni 4119 . 2  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
2 unissb 3989 . 2  |-  ( U. A  C_  B  <->  A. x  e.  A  x  C_  B
)
31, 2bitri 241 1  |-  ( A 
C_  ~P B  <->  A. x  e.  A  x  C_  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wral 2651    C_ wss 3265   ~Pcpw 3744   U.cuni 3959
This theorem is referenced by:  istps5OLD  16914  ustuni  18179  metustfbas  18479  dmvlsiga  24310  1stmbfm  24406  2ndmbfm  24407  dya2iocucvr  24430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-v 2903  df-in 3272  df-ss 3279  df-pw 3746  df-uni 3960
  Copyright terms: Public domain W3C validator