Theorem opeqpr 2880

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 1520	. 2 |- (<.A, B>. = {C, D} <-> {C, D} = <.A, B>.)
2		df-op 2474	. . 3 |- <.A, B>. = {{A}, {A, B}}
3	2	eqeq2i 1528	. 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 2826	. . 3 |- {A} e. V
7		prex 2857	. . 3 |- {A, B} e. V
8	4, 5, 6, 7	preq12b 2548	. 2 |- ({C, D} = {{A}, {A, B}} <-> ((C = {A} /\ D = {A, B}) \/ (C = {A, B} /\ D = {A})))
9	1, 3, 8	3bitri 175	1 |- (<.A, B>. = {C, D} <-> ((C = {A} /\ D = {A, B}) \/ (C = {A, B} /\ D = {A})))

Syntax hints:   <-> wb 144   \/ wo 220   /\ wa 221   = wceq 992   e. wcel 994  Vcvv 1857  {csn 2467  {cpr 2468  <.cop 2469

This theorem is referenced by:  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-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  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  ax-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474

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