Theorem elop 4329

Description: An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypotheses

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

Assertion

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

Proof of Theorem elop

Step	Hyp	Ref	Expression
1		elop.2	. . . 4 |- B e. _V
2		elop.3	. . . 4 |- C e. _V
3	1, 2	dfop 3866	. . 3 |- <. B , C >. = { { B } , { B , C } }
4	3	eleq2i 2298	. 2 |- ( A e. <. B , C >. <-> A e. { { B } , { B , C } } )
5		elop.1	. . 3 |- A e. _V
6	5	elpr 3694	. 2 |- ( A e. { { B } , { B , C } } <-> ( A = { B } \/ A = { B , C } ) )
7	4, 6	bitri 184	1 |- ( A e. <. B , C >. <-> ( A = { B } \/ A = { B , C } ) )

Syntax hints:   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2202   _Vcvv 2803   {csn 3673   {cpr 3674   <.cop 3676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682

This theorem is referenced by:  relop  4886  funopsn  5838  bdop  16591