Theorem elelpwi 3627

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 3624	. . 3 |- ( B e. ~P C -> B C_ C )
2	1	sseld 3191	. 2 |- ( B e. ~P C -> ( A e. B -> A e. C ) )
3	2	impcom 125	1 |- ( ( A e. B /\ B e. ~P C ) -> A e. C )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2175   ~Pcpw 3615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-pw 3617

This theorem is referenced by:  unipw  4260  txdis  14720