Theorem pwel 2754

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 2521	. . 3 |- (A e. B -> A (_ U.B)
2		sspwb 2750	. . 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 2741	. . 3 |- (A e. B -> P~A e. V)
5		elpwg 2401	. . 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 956  Vcvv 1807   (_ wss 2043  P~cpw 2397  U.cuni 2498

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-uni 2499

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