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Theorem pwun 4738
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 4735 . . 3  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
21biantru 505 . 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 4736 . 2  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B ) )
4 eqss 3480 . 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 277 1  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ~P ( A  u.  B )  =  ( ~P A  u.  ~P B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    u. cun 3435    C_ wss 3437   ~Pcpw 3969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-un 3442  df-in 3444  df-ss 3451  df-pw 3971  df-sn 3987  df-pr 3989
This theorem is referenced by: (None)
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