Theorem inpws1 14345

Description: An intersection with a member of a powerset belongs to this powerset.

Assertion

Ref	Expression
inpws1	|- (A e. ~PC -> (A i^i B) e. ~PC)

Proof of Theorem inpws1

Step	Hyp	Ref	Expression
1		inex1g 3454	. 2 |- (A e. ~PC -> (A i^i B) e. _V)
2		elpwg 3038	. . 3 |- ((A i^i B) e. _V -> ((A i^i B) e. ~PC <-> (A i^i B) C_ C))
3		elpwi 3039	. . . 4 |- (A e. ~PC -> A C_ C)
4		ssinss1 2821	. . . 4 |- (A C_ C -> (A i^i B) C_ C)
5	3, 4	syl 12	. . 3 |- (A e. ~PC -> (A i^i B) C_ C)
6	2, 5	syl5bir 227	. 2 |- ((A i^i B) e. _V -> (A e. ~PC -> (A i^i B) e. ~PC))
7	1, 6	mpcom 60	1 |- (A e. ~PC -> (A i^i B) e. ~PC)

Syntax hints:   -> wi 3   e. wcel 1300  _Vcvv 2292   i^i cin 2592   C_ wss 2593  ~Pcpw 3032

This theorem is referenced by:  inpws2 14346  fgsb 14921  fgsb2 14925  inacint 15221

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-ss 2605  df-pw 3035

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