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Theorem pwun 4239
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 4235 . . 3  |-  ( ~P A  u.  ~P B
)  C_  ~P ( A  u.  B )
21biantru 493 . 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 4236 . 2  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B ) )
4 eqss 3136 . 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 270 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 178    \/ wo 359    /\ wa 360    = wceq 1619    u. cun 3092    C_ wss 3094   ~Pcpw 3566
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-un 3099  df-in 3101  df-ss 3108  df-pw 3568  df-sn 3587  df-pr 3588
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