Theorem elop 4276

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 3818	. . 3 |- <. B , C >. = { { B } , { B , C } }
4	3	eleq2i 2272	. 2 |- ( A e. <. B , C >. <-> A e. { { B } , { B , C } } )
5		elop.1	. . 3 |- A e. _V
6	5	elpr 3654	. 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 710   = wceq 1373   e. wcel 2176   _Vcvv 2772   {csn 3633   {cpr 3634   <.cop 3636

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642

This theorem is referenced by:  relop  4829  funopsn  5764  bdop  15848