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Theorem opthreg 6049
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 6036 (via the preleq 6048 step). See df-op 3100 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 3161 . . . 4 |- A e. {A, B}
3 preleq.3 . . . . 5 |- C e. _V
43prid1 3161 . . . 4 |- C e. {C, D}
5 prex 3541 . . . . 5 |- {A, B} e. _V
6 prex 3541 . . . . 5 |- {C, D} e. _V
71, 5, 3, 6preleq 6048 . . . 4 |- (((A e. {A, B} /\ C e. {C, D}) /\ {A, {A, B}} = {C, {C, D}}) -> (A = C /\ {A, B} = {C, D}))
82, 4, 7mpanl12 680 . . 3 |- ({A, {A, B}} = {C, {C, D}} -> (A = C /\ {A, B} = {C, D}))
9 preq1 3142 . . . . . 6 |- (A = C -> {A, B} = {C, B})
109eqeq1d 1961 . . . . 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 3201 . . . . 5 |- ({C, B} = {C, D} -> B = D)
1410, 13syl6bi 233 . . . 4 |- (A = C -> ({A, B} = {C, D} -> B = D))
1514imdistani 689 . . 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 3142 . . . 4 |- (A = C -> {A, {A, B}} = {C, {A, B}})
1817adantr 468 . . 3 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {A, B}})
19 preq12 3144 . . . 4 |- ((A = C /\ B = D) -> {A, B} = {C, D})
2019preq2d 3149 . . 3 |- ((A = C /\ B = D) -> {C, {A, B}} = {C, {C, D}})
2118, 20eqtrd 1985 . 2 |- ((A = C /\ B = D) -> {A, {A, B}} = {C, {C, D}})
2216, 21impbii 188 1 |- ({A, {A, B}} = {C, {C, D}} <-> (A = C /\ B = D))
Colors of variables: wff set class
Syntax hints:   <-> wb 184   /\ wa 377   = wceq 1449   e. wcel 1451  _Vcvv 2358  {cpr 3093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1367  ax-6 1368  ax-7 1369  ax-gen 1370  ax-8 1453  ax-10 1454  ax-11 1455  ax-12 1456  ax-14 1458  ax-17 1465  ax-9 1480  ax-4 1486  ax-16 1664  ax-ext 1935  ax-sep 3469  ax-nul 3478  ax-pr 3538  ax-reg 6036
This theorem depends on definitions:  df-bi 185  df-or 378  df-an 379  df-3an 939  df-ex 1372  df-sb 1626  df-eu 1853  df-mo 1854  df-clab 1941  df-cleq 1946  df-clel 1949  df-ne 2073  df-ral 2166  df-rex 2167  df-v 2360  df-dif 2660  df-un 2662  df-in 2664  df-ss 2666  df-nul 2922  df-sn 3096  df-pr 3097  df-op 3100  df-br 3374  df-opab 3428  df-eprel 3621  df-fr 3662
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