Theorem elop 2859

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 2474	. . 3 |- <.B, C>. = {{B}, {B, C}}
2	1	eleq2i 1581	. 2 |- (A e. <.B, C>. <-> A e. {{B}, {B, C}})
3		elop.1	. . 3 |- A e. V
4	3	elpr 2482	. 2 |- (A e. {{B}, {B, C}} <-> (A = {B} \/ A = {B, C}))
5	2, 4	bitri 171	1 |- (A e. <.B, C>. <-> (A = {B} \/ A = {B, C}))

Syntax hints:   <-> wb 144   \/ wo 220   = wceq 992   e. wcel 994  Vcvv 1857  {csn 2467  {cpr 2468  <.cop 2469

This theorem is referenced by:  opth1 2862  opprc1b 2872  relop 3365

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-v 1858  df-un 2102  df-sn 2470  df-pr 2471  df-op 2474

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