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Theorem pwun 4302
Description: The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of [Enderton] p. 28. (Contributed by NM, 23-Nov-2003.)
Assertion
Ref Expression
pwun  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ~P ( A  u.  B )  =  ( ~P A  u.  ~P B ) )

Proof of Theorem pwun
StepHypRef Expression
1 pwunss 4298 . . 3  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
21biantru 491 . 2  |-  ( ~P ( A  u.  B
)  C_  ( ~P A  u.  ~P B
)  <->  ( ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B )  /\  ( ~P A  u.  ~P B )  C_  ~P ( A  u.  B
) ) )
3 pwssun 4299 . 2  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B ) )
4 eqss 3194 . 2  |-  ( ~P ( A  u.  B
)  =  ( ~P A  u.  ~P B
)  <->  ( ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B )  /\  ( ~P A  u.  ~P B )  C_  ~P ( A  u.  B
) ) )
52, 3, 43bitr4i 268 1  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ~P ( A  u.  B )  =  ( ~P A  u.  ~P B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    u. cun 3150    C_ wss 3152   ~Pcpw 3625
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-pw 3627  df-sn 3646  df-pr 3647
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