Theorem opeqpr 2809

Description: Equivalence for an ordered pair equal to an unordered pair.

Hypotheses

Ref	Expression
opeqpr.3	|- C e. V
opeqpr.4	|- D e. V

Assertion

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

Proof of Theorem opeqpr

Step	Hyp	Ref	Expression
1		eqcom 1480	. 2 |- (<.A, B>. = {C, D} <-> {C, D} = <.A, B>.)
2		df-op 2420	. . 3 |- <.A, B>. = {{A}, {A, B}}
3	2	eqeq2i 1488	. 2 |- ({C, D} = <.A, B>. <-> {C, D} = {{A}, {A, B}})
4		opeqpr.3	. . 3 |- C e. V
5		opeqpr.4	. . 3 |- D e. V
6		snex 2756	. . 3 |- {A} e. V
7		prex 2787	. . 3 |- {A, B} e. V
8	4, 5, 6, 7	preq12b 2487	. 2 |- ({C, D} = {{A}, {A, B}} <-> ((C = {A} /\ D = {A, B}) \/ (C = {A, B} /\ D = {A})))
9	1, 3, 8	3bitr 177	1 |- (<.A, B>. = {C, D} <-> ((C = {A} /\ D = {A, B}) \/ (C = {A, B} /\ D = {A})))

Syntax hints:   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814  {csn 2413  {cpr 2414  <.cop 2415

This theorem is referenced by:  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-11 969  ax-12 970  ax-13 971  ax-14 972  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  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420

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