Theorem neldif 2217

Description: Implication of membership in a class difference.

Assertion

Ref	Expression
neldif	|- ((A e. B /\ -. A e. (B \ C)) -> A e. C)

Proof of Theorem neldif

Step	Hyp	Ref	Expression
1		eldif 2109	. . . . 5 |- (A e. (B \ C) <-> (A e. B /\ -. A e. C))
2	1	biimpri 150	. . . 4 |- ((A e. B /\ -. A e. C) -> A e. (B \ C))
3	2	ex 371	. . 3 |- (A e. B -> (-. A e. C -> A e. (B \ C)))
4	3	con1d 93	. 2 |- (A e. B -> (-. A e. (B \ C) -> A e. C))
5	4	imp 348	1 |- ((A e. B /\ -. A e. (B \ C)) -> A e. C)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 221   e. wcel 994   \ cdif 2096

This theorem is referenced by:  peano5 3241  clsval2 7895

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-v 1858  df-dif 2101

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