Theorem inpws1 10445

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

Assertion

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

Proof of Theorem inpws1

Step	Hyp	Ref	Expression
1		inex1g 2723	. 2 |- (A e. P~C -> (A i^i B) e. V)
2		elpwg 2409	. . 3 |- ((A i^i B) e. V -> ((A i^i B) e. P~C <-> (A i^i B) (_ C))
3		elpwi 2410	. . . 4 |- (A e. P~C -> A (_ C)
4		ssinss1 2240	. . . 4 |- (A (_ C -> (A i^i B) (_ C)
5	3, 4	syl 10	. . 3 |- (A e. P~C -> (A i^i B) (_ C)
6	2, 5	syl5bir 210	. 2 |- ((A i^i B) e. V -> (A e. P~C -> (A i^i B) e. P~C))
7	1, 6	mpcom 49	1 |- (A e. P~C -> (A i^i B) e. P~C)

Syntax hints:   -> wi 3   e. wcel 960  Vcvv 1814   i^i cin 2049   (_ wss 2050  P~cpw 2405

This theorem is referenced by:  inpws2 10446  fgsb 10555  fgsb2 10560

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-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

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-ss 2056  df-pw 2406

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