Theorem elunnel2 39198

Description: A member of a union that is not a member of the second class, is a member of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
elunnel2	|- ( ( A e. ( B u. C ) /\ -. A e. C ) -> A e. B )

Proof of Theorem elunnel2

Step	Hyp	Ref	Expression
1		elun 3753	. . . 4 |- ( A e. ( B u. C ) <-> ( A e. B \/ A e. C ) )
2	1	biimpi 206	. . 3 |- ( A e. ( B u. C ) -> ( A e. B \/ A e. C ) )
3	2	orcomd 403	. 2 |- ( A e. ( B u. C ) -> ( A e. C \/ A e. B ) )
4	3	orcanai 952	1 |- ( ( A e. ( B u. C ) /\ -. A e. C ) -> A e. B )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   e. wcel 1990   u. cun 3572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579

This theorem is referenced by:  limcresiooub  39874  limcresioolb  39875  fourierdlem48  40371  fourierdlem49  40372  fourierdlem101  40424  prsal  40538  isomenndlem  40744  hsphoidmvle2  40799  hsphoidmvle  40800