Theorem opthreg 5709

Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 5695 (via the preleq 5708 step). See df-op 3053 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207.

Hypotheses

Ref	Expression
preleq.1	|- A e. _V
preleq.2	|- B e. _V
preleq.3	|- C e. _V
preleq.4	|- D e. _V

Assertion

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

Proof of Theorem opthreg

Step	Hyp	Ref	Expression
1		preleq.1	. . . . 5 |- A e. _V
2	1	prid1 3106	. . . 4 |- A e. {A, B}
3		preleq.3	. . . . 5 |- C e. _V
4	3	prid1 3106	. . . 4 |- C e. {C, D}
5		prex 3526	. . . . 5 |- {A, B} e. _V
6		prex 3526	. . . . 5 |- {C, D} e. _V
7	1, 5, 3, 6	preleq 5708	. . . 4 |- (((A e. {A, B} /\ C e. {C, D}) /\ {A, {A, B}} = {C, {C, D}}) -> (A = C /\ {A, B} = {C, D}))
8	2, 4, 7	mpanl12 773	. . 3 |- ({A, {A, B}} = {C, {C, D}} -> (A = C /\ {A, B} = {C, D}))
9		preq1 3098	. . . . . 6 |- (A = C -> {A, B} = {C, B})
10	9	eqeq1d 1892	. . . . 5 |- (A = C -> ({A, B} = {C, D} <-> {C, B} = {C, D}))
11		preleq.2	. . . . . 6 |- B e. _V
12		preleq.4	. . . . . 6 |- D e. _V
13	11, 12	preqr2 3153	. . . . 5 |- ({C, B} = {C, D} -> B = D)
14	10, 13	syl6bi 231	. . . 4 |- (A = C -> ({A, B} = {C, D} -> B = D))
15	14	imdistani 491	. . 3 |- ((A = C /\ {A, B} = {C, D}) -> (A = C /\ B = D))
16	8, 15	syl 12	. 2 |- ({A, {A, B}} = {C, {C, D}} -> (A = C /\ B = D))
17		preq1 3098	. . . 4 |- (A = C -> {A, {A, B}} = {C, {A, B}})
18	17	adantr 425	. . 3 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {A, B}})
19		preq2 3099	. . . . 5 |- (B = D -> {C, B} = {C, D})
20	9, 19	sylan9eq 1948	. . . 4 |- ((A = C /\ B = D) -> {A, B} = {C, D})
21		preq2 3099	. . . 4 |- ({A, B} = {C, D} -> {C, {A, B}} = {C, {C, D}})
22	20, 21	syl 12	. . 3 |- ((A = C /\ B = D) -> {C, {A, B}} = {C, {C, D}})
23	18, 22	eqtrd 1925	. 2 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {C, D}})
24	16, 23	impbii 174	1 |- ({A, {A, B}} = {C, {C, D}} <-> (A = C /\ B = D))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  {cpr 3045

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-14 1312  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  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-eprel 3583  df-fr 3625

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