Theorem pwun 3580

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.

Assertion

Ref	Expression
pwun	|- ((A C_ B \/ B C_ A) <-> ~P(A u. B) = (~PA u. ~PB))

Proof of Theorem pwun

Step	Hyp	Ref	Expression
1		pwunss 3577	. . 3 |- (~PA u. ~PB) C_ ~P(A u. B)
2	1	biantru 793	. 2 |- (~P(A u. B) C_ (~PA u. ~PB) <-> (~P(A u. B) C_ (~PA u. ~PB) /\ (~PA u. ~PB) C_ ~P(A u. B)))
3		pwssun 3578	. 2 |- ((A C_ B \/ B C_ A) <-> ~P(A u. B) C_ (~PA u. ~PB))
4		eqss 2631	. 2 |- (~P(A u. B) = (~PA u. ~PB) <-> (~P(A u. B) C_ (~PA u. ~PB) /\ (~PA u. ~PB) C_ ~P(A u. B)))
5	2, 3, 4	3bitr4i 200	1 |- ((A C_ B \/ B C_ A) <-> ~P(A u. B) = (~PA u. ~PB))

Syntax hints:   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   u. cun 2591   C_ wss 2593  ~Pcpw 3032

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-in 2603  df-ss 2605  df-pw 3035  df-sn 3049  df-pr 3050

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