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Theorem opthreg 6115
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 6102 (via the preleq 6114 step). See df-op 3241 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 3298 . . . 4 |- A e. {A, B}
3 preleq.3 . . . . 5 |- C e. _V
43prid1 3298 . . . 4 |- C e. {C, D}
5 prex 3682 . . . . 5 |- {A, B} e. _V
6 prex 3682 . . . . 5 |- {C, D} e. _V
71, 5, 3, 6preleq 6114 . . . 4 |- (((A e. {A, B} /\ C e. {C, D}) /\ {A, {A, B}} = {C, {C, D}}) -> (A = C /\ {A, B} = {C, D}))
82, 4, 7mpanl12 768 . . 3 |- ({A, {A, B}} = {C, {C, D}} -> (A = C /\ {A, B} = {C, D}))
9 preq1 3283 . . . . . 6 |- (A = C -> {A, B} = {C, B})
109eqeq1d 2100 . . . . 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 3338 . . . . 5 |- ({C, B} = {C, D} -> B = D)
1410, 13syl6bi 264 . . . 4 |- (A = C -> ({A, B} = {C, D} -> B = D))
1514imdistani 781 . . 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 3283 . . . 4 |- (A = C -> {A, {A, B}} = {C, {A, B}})
1817adantr 516 . . 3 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {A, B}})
19 preq12 3285 . . . 4 |- ((A = C /\ B = D) -> {A, B} = {C, D})
2019preq2d 3290 . . 3 |- ((A = C /\ B = D) -> {C, {A, B}} = {C, {C, D}})
2118, 20eqtrd 2124 . 2 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {C, D}})
2216, 21impbii 213 1 |- ({A, {A, B}} = {C, {C, D}} <-> (A = C /\ B = D))
Colors of variables: wff set class
Syntax hints:   <-> wb 209   /\ wa 418   = wceq 1592   e. wcel 1594  _Vcvv 2499  {cpr 3234
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1516  ax-6 1517  ax-7 1518  ax-gen 1519  ax-8 1596  ax-10 1597  ax-11 1598  ax-12 1599  ax-14 1601  ax-17 1608  ax-9 1620  ax-4 1626  ax-16 1803  ax-ext 2074  ax-sep 3609  ax-nul 3619  ax-pr 3679  ax-reg 6102
This theorem depends on definitions:  df-bi 210  df-or 419  df-an 420  df-3an 1039  df-ex 1521  df-sb 1765  df-eu 1992  df-mo 1993  df-clab 2080  df-cleq 2085  df-clel 2088  df-ne 2220  df-ral 2314  df-rex 2315  df-v 2501  df-dif 2804  df-un 2806  df-in 2808  df-ss 2810  df-nul 3066  df-sn 3237  df-pr 3238  df-op 3241  df-br 3510  df-opab 3568  df-eprel 3762  df-fr 3800
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