Theorem opthreg 4749

Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 4736 (via the preleq 4748 step). See df-op 2474 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 2513	. . . 4 |- A e. {A, B}
3		preleq.3	. . . . 5 |- C e. V
4	3	prid1 2513	. . . 4 |- C e. {C, D}
5		prex 2857	. . . . 5 |- {A, B} e. V
6		prex 2857	. . . . 5 |- {C, D} e. V
7	1, 5, 3, 6	preleq 4748	. . . 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 712	. . 3 |- ({A, {A, B}} = {C, {C, D}} -> (A = C /\ {A, B} = {C, D}))
9		preq1 2509	. . . . . 6 |- (A = C -> {A, B} = {C, B})
10	9	eqeq1d 1526	. . . . 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 2547	. . . . 5 |- ({C, B} = {C, D} -> B = D)
14	10, 13	syl6bi 212	. . . 4 |- (A = C -> ({A, B} = {C, D} -> B = D))
15	14	imdistani 445	. . 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 2509	. . . 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 2510	. . . . 5 |- (B = D -> {C, B} = {C, D})
20	9, 19	sylan9eq 1570	. . . 4 |- ((A = C /\ B = D) -> {A, B} = {C, D})
21		preq2 2510	. . . 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 1550	. 2 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {C, D}})
24	16, 23	impbii 155	1 |- ({A, {A, B}} = {C, {C, D}} <-> (A = C /\ B = D))

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

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  ax-reg 4736

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  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-ral 1695  df-rex 1696  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  df-br 2693  df-opab 2741  df-eprel 2910  df-fr 2947

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