Theorem opthreg 4604

Description: Theorem for alternate representation of ordered pairs, requiring Regularity. 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	pri1 2450	. . . 4 |- A e. {A, B}
3		preleq.3	. . . . 5 |- C e. V
4	3	pri1 2450	. . . 4 |- C e. {C, D}
5		prex 2781	. . . . 5 |- {A, B} e. V
6		prex 2781	. . . . 5 |- {C, D} e. V
7	1, 5, 3, 6	preleq 4603	. . . 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 708	. . 3 |- ({A, {A, B}} = {C, {C, D}} -> (A = C /\ {A, B} = {C, D}))
9		preq1 2448	. . . . . 6 |- (A = C -> {A, B} = {C, B})
10	9	eqeq1d 1483	. . . . 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 2482	. . . . 5 |- ({C, B} = {C, D} -> B = D)
14	10, 13	syl6bi 214	. . . 4 |- (A = C -> ({A, B} = {C, D} -> B = D))
15	14	imdistani 443	. . 3 |- ((A = C /\ {A, B} = {C, D}) -> (A = C /\ B = D))
16	8, 15	syl 10	. 2 |- ({A, {A, B}} = {C, {C, D}} -> (A = C /\ B = D))
17		preq1 2448	. . . 4 |- (A = C -> {A, {A, B}} = {C, {A, B}})
18	17	adantr 389	. . 3 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {A, B}})
19		preq2 2449	. . . . 5 |- (B = D -> {C, B} = {C, D})
20	9, 19	sylan9eq 1527	. . . 4 |- ((A = C /\ B = D) -> {A, B} = {C, D})
21		preq2 2449	. . . 4 |- ({A, B} = {C, D} -> {C, {A, B}} = {C, {C, D}})
22	20, 21	syl 10	. . 3 |- ((A = C /\ B = D) -> {C, {A, B}} = {C, {C, D}})
23	18, 22	eqtrd 1507	. 2 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {C, D}})
24	16, 23	impbi 157	1 |- ({A, {A, B}} = {C, {C, D}} <-> (A = C /\ B = D))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  {cpr 2410

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-11 967  ax-12 968  ax-13 969  ax-14 970  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  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-reg 4593

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-eprel 2832  df-fr 2917

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