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Theorem opthreg 6057
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 6044 (via the preleq 6056 step). See df-op 3088 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 3151 . . . 4 |- A e. {A, B}
3 preleq.3 . . . . 5 |- C e. _V
43prid1 3151 . . . 4 |- C e. {C, D}
5 prex 3531 . . . . 5 |- {A, B} e. _V
6 prex 3531 . . . . 5 |- {C, D} e. _V
71, 5, 3, 6preleq 6056 . . . 4 |- (((A e. {A, B} /\ C e. {C, D}) /\ {A, {A, B}} = {C, {C, D}}) -> (A = C /\ {A, B} = {C, D}))
82, 4, 7mpanl12 666 . . 3 |- ({A, {A, B}} = {C, {C, D}} -> (A = C /\ {A, B} = {C, D}))
9 preq1 3132 . . . . . 6 |- (A = C -> {A, B} = {C, B})
109eqeq1d 1946 . . . . 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 3191 . . . . 5 |- ({C, B} = {C, D} -> B = D)
1410, 13syl6bi 218 . . . 4 |- (A = C -> ({A, B} = {C, D} -> B = D))
1514imdistani 675 . . 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 3132 . . . 4 |- (A = C -> {A, {A, B}} = {C, {A, B}})
1817adantr 452 . . 3 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {A, B}})
19 preq12 3134 . . . 4 |- ((A = C /\ B = D) -> {A, B} = {C, D})
2019preq2d 3139 . . 3 |- ((A = C /\ B = D) -> {C, {A, B}} = {C, {C, D}})
2118, 20eqtrd 1970 . 2 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {C, D}})
2216, 21impbii 178 1 |- ({A, {A, B}} = {C, {C, D}} <-> (A = C /\ B = D))
Colors of variables: wff set class
Syntax hints:   <-> wb 174   /\ wa 361   = wceq 1434   e. wcel 1436  _Vcvv 2343  {cpr 3080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1351  ax-6 1352  ax-7 1353  ax-gen 1354  ax-8 1438  ax-10 1439  ax-11 1440  ax-12 1441  ax-14 1443  ax-17 1450  ax-9 1465  ax-4 1471  ax-16 1649  ax-ext 1920  ax-sep 3459  ax-nul 3468  ax-pr 3528  ax-reg 6044
This theorem depends on definitions:  df-bi 175  df-or 362  df-an 363  df-3an 923  df-ex 1356  df-sb 1611  df-eu 1838  df-mo 1839  df-clab 1926  df-cleq 1931  df-clel 1934  df-ne 2058  df-ral 2151  df-rex 2152  df-v 2345  df-dif 2645  df-un 2647  df-in 2649  df-ss 2651  df-nul 2907  df-sn 3085  df-pr 3086  df-op 3088  df-br 3364  df-opab 3418  df-eprel 3613  df-fr 3654
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