Theorem elelpwi 3370

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 3368	. . 3 |- ( B e. ~P C -> B C_ C )
2	1	sseld 2944	. 2 |- ( B e. ~P C -> ( A e. B -> A e. C ) )
3	2	impcom 116	1 |- ( ( A e. B /\ B e. ~P C ) -> A e. C )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   ~Pcpw 3359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361

This theorem is referenced by:  unipw  3953