Theorem elpreqprb 4397

Description: A set is an element of an unordered pair iff there is another (maybe the same) set which is an element of the unordered pair. (Proposed by BJ, 8-Dec-2020.) (Contributed by AV, 9-Dec-2020.)

Assertion

Ref	Expression
elpreqprb	|- ( A e. V -> ( A e. { B , C } <-> E. x { B , C } = { A , x } ) )

Distinct variable groups:   x, A   x, B   x, C   x, V

Proof of Theorem elpreqprb

Step	Hyp	Ref	Expression
1		elpreqpr 4396	. 2 |- ( A e. { B , C } -> E. x { B , C } = { A , x } )
2		prid1g 4295	. . . 4 |- ( A e. V -> A e. { A , x } )
3		eleq2 2690	. . . 4 |- ( { B , C } = { A , x } -> ( A e. { B , C } <-> A e. { A , x } ) )
4	2, 3	syl5ibrcom 237	. . 3 |- ( A e. V -> ( { B , C } = { A , x } -> A e. { B , C } ) )
5	4	exlimdv 1861	. 2 |- ( A e. V -> ( E. x { B , C } = { A , x } -> A e. { B , C } ) )
6	1, 5	impbid2 216	1 |- ( A e. V -> ( A e. { B , C } <-> E. x { B , C } = { A , x } ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   {cpr 4179

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-un 3579  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)