Theorem elop 3528

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 3053	. . 3 |- <.B, C>. = {{B}, {B, C}}
2	1	eleq2i 1961	. 2 |- (A e. <.B, C>. <-> A e. {{B}, {B, C}})
3		elop.1	. . 3 |- A e. _V
4	3	elpr 3061	. 2 |- (A e. {{B}, {B, C}} <-> (A = {B} \/ A = {B, C}))
5	2, 4	bitri 190	1 |- (A e. <.B, C>. <-> (A = {B} \/ A = {B, C}))

Syntax hints:   <-> wb 163   \/ wo 239   = wceq 1298   e. wcel 1300  _Vcvv 2292  {csn 3044  {cpr 3045  <.cop 3046

This theorem is referenced by:  opth1 3531  opprc1b 3542  relop 4113

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-op 3053

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