Theorem eqoreldif 4004

Description: An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020.)

Assertion

Ref	Expression
eqoreldif	|- ( B e. C -> ( A e. C <-> ( A = B \/ A e. ( C \ { B } ) ) ) )

Proof of Theorem eqoreldif

Step	Hyp	Ref	Expression
1		orc 392	. . . . 5 |- ( A = B -> ( A = B \/ A e. ( C \ { B } ) ) )
2	1	a1d 25	. . . 4 |- ( A = B -> ( ( B e. C /\ A e. C ) -> ( A = B \/ A e. ( C \ { B } ) ) ) )
3		simprr 774	. . . . . . 7 |- ( ( -. A = B /\ ( B e. C /\ A e. C ) ) -> A e. C )
4		elsni 3985	. . . . . . . . . 10 |- ( A e. { B } -> A = B )
5	4	a1i 11	. . . . . . . . 9 |- ( ( B e. C /\ A e. C ) -> ( A e. { B } -> A = B ) )
6	5	con3d 140	. . . . . . . 8 |- ( ( B e. C /\ A e. C ) -> ( -. A = B -> -. A e. { B } ) )
7	6	impcom 437	. . . . . . 7 |- ( ( -. A = B /\ ( B e. C /\ A e. C ) ) -> -. A e. { B } )
8	3, 7	eldifd 3401	. . . . . 6 |- ( ( -. A = B /\ ( B e. C /\ A e. C ) ) -> A e. ( C \ { B } ) )
9	8	olcd 400	. . . . 5 |- ( ( -. A = B /\ ( B e. C /\ A e. C ) ) -> ( A = B \/ A e. ( C \ { B } ) ) )
10	9	ex 441	. . . 4 |- ( -. A = B -> ( ( B e. C /\ A e. C ) -> ( A = B \/ A e. ( C \ { B } ) ) ) )
11	2, 10	pm2.61i 169	. . 3 |- ( ( B e. C /\ A e. C ) -> ( A = B \/ A e. ( C \ { B } ) ) )
12	11	ex 441	. 2 |- ( B e. C -> ( A e. C -> ( A = B \/ A e. ( C \ { B } ) ) ) )
13		eleq1 2537	. . . . 5 |- ( A = B -> ( A e. C <-> B e. C ) )
14	13	biimprd 231	. . . 4 |- ( A = B -> ( B e. C -> A e. C ) )
15		eldifi 3544	. . . . 5 |- ( A e. ( C \ { B } ) -> A e. C )
16	15	a1d 25	. . . 4 |- ( A e. ( C \ { B } ) -> ( B e. C -> A e. C ) )
17	14, 16	jaoi 386	. . 3 |- ( ( A = B \/ A e. ( C \ { B } ) ) -> ( B e. C -> A e. C ) )
18	17	com12 31	. 2 |- ( B e. C -> ( ( A = B \/ A e. ( C \ { B } ) ) -> A e. C ) )
19	12, 18	impbid 195	1 |- ( B e. C -> ( A e. C <-> ( A = B \/ A e. ( C \ { B } ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   = wceq 1452   e. wcel 1904   \ cdif 3387   {csn 3959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-dif 3393  df-sn 3960

This theorem is referenced by:  lcmfunsnlem2  14692