Theorem elopOLD 4936

Description: Obsolete version of elop 4935, with one extraneous hypothesis. Obsolete as of 25-Dec-2020 . (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
elop.1	|- B e. _V
elop.2	|- C e. _V
elopOLD.3	|- A e. _V

Assertion

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

Proof of Theorem elopOLD

Step	Hyp	Ref	Expression
1		elop.1	. . . 4 |- B e. _V
2		elop.2	. . . 4 |- C e. _V
3	1, 2	dfop 4401	. . 3 |- <. B , C >. = { { B } , { B , C } }
4	3	eleq2i 2693	. 2 |- ( A e. <. B , C >. <-> A e. { { B } , { B , C } } )
5		elopOLD.3	. . 3 |- A e. _V
6	5	elpr 4198	. 2 |- ( A e. { { B } , { B , C } } <-> ( A = { B } \/ A = { B , C } ) )
7	4, 6	bitri 264	1 |- ( A e. <. B , C >. <-> ( A = { B } \/ A = { B , C } ) )

Syntax hints:   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   {cpr 4179   <.cop 4183

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-3an 1039  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-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by: (None)