Theorem pwun 2829

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 (_ B \/ B (_ A) <-> P~(A u. B) = (P~A u. P~B))

Proof of Theorem pwun

Step	Hyp	Ref	Expression
1		pwunss 2826	. . 3 |- (P~A u. P~B) (_ P~(A u. B)
2	1	biantru 724	. 2 |- (P~(A u. B) (_ (P~A u. P~B) <-> (P~(A u. B) (_ (P~A u. P~B) /\ (P~A u. P~B) (_ P~(A u. B)))
3		pwssun 2827	. 2 |- ((A (_ B \/ B (_ A) <-> P~(A u. B) (_ (P~A u. P~B))
4		eqss 2077	. 2 |- (P~(A u. B) = (P~A u. P~B) <-> (P~(A u. B) (_ (P~A u. P~B) /\ (P~A u. P~B) (_ P~(A u. B)))
5	2, 3, 4	3bitr4 183	1 |- ((A (_ B \/ B (_ A) <-> P~(A u. B) = (P~A u. P~B))

Syntax hints:   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956   u. cun 2045   (_ wss 2047  P~cpw 2401

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-in 2051  df-ss 2053  df-pw 2402  df-sn 2412  df-pr 2413

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