Theorem pwun 2835

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 2832	. . 3 |- (P~A u. P~B) (_ P~(A u. B)
2	1	biantru 726	. 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 2833	. 2 |- ((A (_ B \/ B (_ A) <-> P~(A u. B) (_ (P~A u. P~B))
4		eqss 2080	. 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 958   u. cun 2048   (_ wss 2050  P~cpw 2405

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-in 2054  df-ss 2056  df-pw 2406  df-sn 2416  df-pr 2417

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