Theorem eqoreldifOLD 4226

Description: Obsolete proof of eqoreldif 4225 as of 23-Jul-2021. (Contributed by AV, 25-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem eqoreldifOLD

Step	Hyp	Ref	Expression
1		orc 400	. . . . 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 796	. . . . . . 7 |- ( ( -. A = B /\ ( B e. C /\ A e. C ) ) -> A e. C )
4		elsni 4194	. . . . . . . . . 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 148	. . . . . . . 8 |- ( ( B e. C /\ A e. C ) -> ( -. A = B -> -. A e. { B } ) )
7	6	impcom 446	. . . . . . 7 |- ( ( -. A = B /\ ( B e. C /\ A e. C ) ) -> -. A e. { B } )
8	3, 7	eldifd 3585	. . . . . 6 |- ( ( -. A = B /\ ( B e. C /\ A e. C ) ) -> A e. ( C \ { B } ) )
9	8	olcd 408	. . . . 5 |- ( ( -. A = B /\ ( B e. C /\ A e. C ) ) -> ( A = B \/ A e. ( C \ { B } ) ) )
10	9	ex 450	. . . 4 |- ( -. A = B -> ( ( B e. C /\ A e. C ) -> ( A = B \/ A e. ( C \ { B } ) ) ) )
11	2, 10	pm2.61i 176	. . 3 |- ( ( B e. C /\ A e. C ) -> ( A = B \/ A e. ( C \ { B } ) ) )
12	11	ex 450	. 2 |- ( B e. C -> ( A e. C -> ( A = B \/ A e. ( C \ { B } ) ) ) )
13		eleq1 2689	. . . . 5 |- ( A = B -> ( A e. C <-> B e. C ) )
14	13	biimprd 238	. . . 4 |- ( A = B -> ( B e. C -> A e. C ) )
15		eldifi 3732	. . . . 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 394	. . 3 |- ( ( A = B \/ A e. ( C \ { B } ) ) -> ( B e. C -> A e. C ) )
18	17	com12 32	. 2 |- ( B e. C -> ( ( A = B \/ A e. ( C \ { B } ) ) -> A e. C ) )
19	12, 18	impbid 202	1 |- ( B e. C -> ( A e. C <-> ( A = B \/ A e. ( C \ { B } ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   {csn 4177

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-dif 3577  df-sn 4178

This theorem is referenced by: (None)