Theorem elelpwi 3469

Description: If A belongs to a part of C then A belongs to C. (Contributed by FL, 3-Aug-2009.)

Assertion

Ref	Expression
elelpwi	|- ( ( A e. B /\ B e. ~P C ) -> A e. C )

Proof of Theorem elelpwi

Step	Hyp	Ref	Expression
1		elpwi 3466	. . 3 |- ( B e. ~P C -> B C_ C )
2	1	sseld 3046	. 2 |- ( B e. ~P C -> ( A e. B -> A e. C ) )
3	2	impcom 124	1 |- ( ( A e. B /\ B e. ~P C ) -> A e. C )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1448   ~Pcpw 3457

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459

This theorem is referenced by:  unipw  4077  txdis  12227