Theorem pwel 2759

Description: Membership of a power class. Exercise 10 of [Enderton] p. 26.

Assertion

Ref	Expression
pwel	|- (A e. B -> P~A e. P~P~U.B)

Proof of Theorem pwel

Step	Hyp	Ref	Expression
1		elssuni 2526	. . 3 |- (A e. B -> A (_ U.B)
2		sspwb 2755	. . 3 |- (A (_ U.B <-> P~A (_ P~U.B)
3	1, 2	sylib 198	. 2 |- (A e. B -> P~A (_ P~U.B)
4		pwexg 2746	. . 3 |- (A e. B -> P~A e. V)
5		elpwg 2405	. . 3 |- (P~A e. V -> (P~A e. P~P~U.B <-> P~A (_ P~U.B))
6	4, 5	syl 10	. 2 |- (A e. B -> (P~A e. P~P~U.B <-> P~A (_ P~U.B))
7	3, 6	mpbird 196	1 |- (A e. B -> P~A e. P~P~U.B)

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 958  Vcvv 1811   (_ wss 2047  P~cpw 2401  U.cuni 2503

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-11 967  ax-12 968  ax-13 969  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-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-uni 2504

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