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Theorem pwun 4451
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 4448 . . 3  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
21biantru 492 . 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 4449 . 2  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B ) )
4 eqss 3323 . 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 269 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 177    \/ wo 358    /\ wa 359    = wceq 1649    u. cun 3278    C_ wss 3280   ~Pcpw 3759
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-pr 4363
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285  df-in 3287  df-ss 3294  df-pw 3761  df-sn 3780  df-pr 3781
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