Theorem elop 2839

Description: An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15.

Hypothesis

Ref	Expression
elop.1	|- A e. V

Assertion

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

Proof of Theorem elop

Step	Hyp	Ref	Expression
1		df-op 2468	. . 3 |- <.B, C>. = {{B}, {B, C}}
2	1	eleq2i 1585	. 2 |- (A e. <.B, C>. <-> A e. {{B}, {B, C}})
3		elop.1	. . 3 |- A e. V
4	3	elpr 2476	. 2 |- (A e. {{B}, {B, C}} <-> (A = {B} \/ A = {B, C}))
5	2, 4	bitri 180	1 |- (A e. <.B, C>. <-> (A = {B} \/ A = {B, C}))

Syntax hints:   <-> wb 153   \/ wo 229   = wceq 997   e. wcel 999  Vcvv 1858  {csn 2461  {cpr 2462  <.cop 2463

This theorem is referenced by:  opth1 2842  opprc1b 2852  relop 3332

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-v 1859  df-un 2101  df-sn 2464  df-pr 2465  df-op 2468

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