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Theorem pwun 4747
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 4744 . . 3  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
21biantru 513 . 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 4745 . 2  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B ) )
4 eqss 3433 . 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 285 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 189    \/ wo 375    /\ wa 376    = wceq 1452    u. cun 3388    C_ wss 3390   ~Pcpw 3942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-un 3395  df-in 3397  df-ss 3404  df-pw 3944  df-sn 3960  df-pr 3962
This theorem is referenced by: (None)
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