Theorem elop 4264

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 3807	. . 3 |- <. B , C >. = { { B } , { B , C } }
4	3	eleq2i 2263	. 2 |- ( A e. <. B , C >. <-> A e. { { B } , { B , C } } )
5		elop.1	. . 3 |- A e. _V
6	5	elpr 3643	. 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 709   = wceq 1364   e. wcel 2167   _Vcvv 2763   {csn 3622   {cpr 3623   <.cop 3625

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631

This theorem is referenced by:  relop  4816  bdop  15521