Theorem elop 2783

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 2416	. . 3 |- <.B, C>. = {{B}, {B, C}}
2	1	eleq2i 1538	. 2 |- (A e. <.B, C>. <-> A e. {{B}, {B, C}})
3		elop.1	. . 3 |- A e. V
4	3	elpr 2424	. 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 956   e. wcel 958  Vcvv 1811  {csn 2409  {cpr 2410  <.cop 2411

This theorem is referenced by:  opth1 2786  opprc1b 2796  relop 3275

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416

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