Theorem elop 4260

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 3803	. . 3 |- <. B , C >. = { { B } , { B , C } }
4	3	eleq2i 2260	. 2 |- ( A e. <. B , C >. <-> A e. { { B } , { B , C } } )
5		elop.1	. . 3 |- A e. _V
6	5	elpr 3639	. 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 2164   _Vcvv 2760   {csn 3618   {cpr 3619   <.cop 3621

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627

This theorem is referenced by:  relop  4812  bdop  15367