Theorem elop 2789

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 2420	. . 3 |- <.B, C>. = {{B}, {B, C}}
2	1	eleq2i 1541	. 2 |- (A e. <.B, C>. <-> A e. {{B}, {B, C}})
3		elop.1	. . 3 |- A e. V
4	3	elpr 2428	. 2 |- (A e. {{B}, {B, C}} <-> (A = {B} \/ A = {B, C}))
5	2, 4	bitr 173	1 |- (A e. <.B, C>. <-> (A = {B} \/ A = {B, C}))

Syntax hints:   <-> wb 146   \/ wo 222   = wceq 958   e. wcel 960  Vcvv 1814  {csn 2413  {cpr 2414  <.cop 2415

This theorem is referenced by:  opth1 2792  opprc1b 2802  relop 3281

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420

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