Theorem elelpwi 3633

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 3630	. . 3 |- ( B e. ~P C -> B C_ C )
2	1	sseld 3196	. 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 2177   ~Pcpw 3621

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-ss 3183  df-pw 3623

This theorem is referenced by:  unipw  4274  txdis  14834  uhgredgrnv  15814