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Theorem opthreg 6050
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 6037 (via the preleq 6049 step). See df-op 3106 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
StepHypRef Expression
1 preleq.1 . . . . 5 |- A e. _V
21prid1 3167 . . . 4 |- A e. {A, B}
3 preleq.3 . . . . 5 |- C e. _V
43prid1 3167 . . . 4 |- C e. {C, D}
5 prex 3547 . . . . 5 |- {A, B} e. _V
6 prex 3547 . . . . 5 |- {C, D} e. _V
71, 5, 3, 6preleq 6049 . . . 4 |- (((A e. {A, B} /\ C e. {C, D}) /\ {A, {A, B}} = {C, {C, D}}) -> (A = C /\ {A, B} = {C, D}))
82, 4, 7mpanl12 689 . . 3 |- ({A, {A, B}} = {C, {C, D}} -> (A = C /\ {A, B} = {C, D}))
9 preq1 3148 . . . . . 6 |- (A = C -> {A, B} = {C, B})
109eqeq1d 1969 . . . . 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
1311, 12preqr2 3207 . . . . 5 |- ({C, B} = {C, D} -> B = D)
1410, 13syl6bi 238 . . . 4 |- (A = C -> ({A, B} = {C, D} -> B = D))
1514imdistani 698 . . 3 |- ((A = C /\ {A, B} = {C, D}) -> (A = C /\ B = D))
168, 15syl 14 . 2 |- ({A, {A, B}} = {C, {C, D}} -> (A = C /\ B = D))
17 preq1 3148 . . . 4 |- (A = C -> {A, {A, B}} = {C, {A, B}})
1817adantr 474 . . 3 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {A, B}})
19 preq12 3150 . . . 4 |- ((A = C /\ B = D) -> {A, B} = {C, D})
2019preq2d 3155 . . 3 |- ((A = C /\ B = D) -> {C, {A, B}} = {C, {C, D}})
2118, 20eqtrd 1993 . 2 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {C, D}})
2216, 21impbii 193 1 |- ({A, {A, B}} = {C, {C, D}} <-> (A = C /\ B = D))
Colors of variables: wff set class
Syntax hints:   <-> wb 189   /\ wa 382   = wceq 1457   e. wcel 1459  _Vcvv 2366  {cpr 3099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1376  ax-6 1377  ax-7 1378  ax-gen 1379  ax-8 1461  ax-10 1462  ax-11 1463  ax-12 1464  ax-14 1466  ax-17 1473  ax-9 1488  ax-4 1494  ax-16 1672  ax-ext 1943  ax-sep 3475  ax-nul 3484  ax-pr 3544  ax-reg 6037
This theorem depends on definitions:  df-bi 190  df-or 383  df-an 384  df-3an 948  df-ex 1381  df-sb 1634  df-eu 1861  df-mo 1862  df-clab 1949  df-cleq 1954  df-clel 1957  df-ne 2081  df-ral 2174  df-rex 2175  df-v 2368  df-dif 2666  df-un 2668  df-in 2670  df-ss 2672  df-nul 2928  df-sn 3102  df-pr 3103  df-op 3106  df-br 3380  df-opab 3434  df-eprel 3627  df-fr 3668
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