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Theorem opthreg 6098
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 6085 (via the preleq 6097 step). See df-op 3224 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 3281 . . . 4 |- A e. {A, B}
3 preleq.3 . . . . 5 |- C e. _V
43prid1 3281 . . . 4 |- C e. {C, D}
5 prex 3665 . . . . 5 |- {A, B} e. _V
6 prex 3665 . . . . 5 |- {C, D} e. _V
71, 5, 3, 6preleq 6097 . . . 4 |- (((A e. {A, B} /\ C e. {C, D}) /\ {A, {A, B}} = {C, {C, D}}) -> (A = C /\ {A, B} = {C, D}))
82, 4, 7mpanl12 751 . . 3 |- ({A, {A, B}} = {C, {C, D}} -> (A = C /\ {A, B} = {C, D}))
9 preq1 3266 . . . . . 6 |- (A = C -> {A, B} = {C, B})
109eqeq1d 2083 . . . . 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 3321 . . . . 5 |- ({C, B} = {C, D} -> B = D)
1410, 13syl6bi 258 . . . 4 |- (A = C -> ({A, B} = {C, D} -> B = D))
1514imdistani 764 . . 3 |- ((A = C /\ {A, B} = {C, D}) -> (A = C /\ B = D))
168, 15syl 13 . 2 |- ({A, {A, B}} = {C, {C, D}} -> (A = C /\ B = D))
17 preq1 3266 . . . 4 |- (A = C -> {A, {A, B}} = {C, {A, B}})
1817adantr 505 . . 3 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {A, B}})
19 preq12 3268 . . . 4 |- ((A = C /\ B = D) -> {A, B} = {C, D})
2019preq2d 3273 . . 3 |- ((A = C /\ B = D) -> {C, {A, B}} = {C, {C, D}})
2118, 20eqtrd 2107 . 2 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {C, D}})
2216, 21impbii 207 1 |- ({A, {A, B}} = {C, {C, D}} <-> (A = C /\ B = D))
Colors of variables: wff set class
Syntax hints:   <-> wb 203   /\ wa 412   = wceq 1573   e. wcel 1575  _Vcvv 2482  {cpr 3217
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1497  ax-6 1498  ax-7 1499  ax-gen 1500  ax-8 1577  ax-10 1578  ax-11 1579  ax-12 1580  ax-14 1582  ax-17 1589  ax-9 1603  ax-4 1609  ax-16 1786  ax-ext 2057  ax-sep 3592  ax-nul 3602  ax-pr 3662  ax-reg 6085
This theorem depends on definitions:  df-bi 204  df-or 413  df-an 414  df-3an 1020  df-ex 1502  df-sb 1748  df-eu 1975  df-mo 1976  df-clab 2063  df-cleq 2068  df-clel 2071  df-ne 2203  df-ral 2297  df-rex 2298  df-v 2484  df-dif 2787  df-un 2789  df-in 2791  df-ss 2793  df-nul 3049  df-sn 3220  df-pr 3221  df-op 3224  df-br 3493  df-opab 3551  df-eprel 3745  df-fr 3783
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