Theorem pwunss 2826

Description: The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235.

Assertion

Ref	Expression
pwunss	|- (P~A u. P~B) (_ P~(A u. B)

Proof of Theorem pwunss

Step	Hyp	Ref	Expression
1		ssun 2206	. . 3 |- ((x (_ A \/ x (_ B) -> x (_ (A u. B))
2		elun 2173	. . . 4 |- (x e. (P~A u. P~B) <-> (x e. P~A \/ x e. P~B))
3		visset 1813	. . . . . 6 |- x e. V
4	3	elpw 2404	. . . . 5 |- (x e. P~A <-> x (_ A)
5	3	elpw 2404	. . . . 5 |- (x e. P~B <-> x (_ B)
6	4, 5	orbi12i 257	. . . 4 |- ((x e. P~A \/ x e. P~B) <-> (x (_ A \/ x (_ B))
7	2, 6	bitr 173	. . 3 |- (x e. (P~A u. P~B) <-> (x (_ A \/ x (_ B))
8	3	elpw 2404	. . 3 |- (x e. P~(A u. B) <-> x (_ (A u. B))
9	1, 7, 8	3imtr4 219	. 2 |- (x e. (P~A u. P~B) -> x e. P~(A u. B))
10	9	ssriv 2069	1 |- (P~A u. P~B) (_ P~(A u. B)

Syntax hints:   \/ wo 222   e. wcel 958   u. cun 2045   (_ wss 2047  P~cpw 2401

This theorem is referenced by:  pwun 2829

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-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

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

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