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Theorem opthreg 4218
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4204 (via the preleq 4217 step). See df-op 3359 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
Hypotheses
Ref Expression
preleq.1 A V
preleq.2 B V
preleq.3 𝐶 V
preleq.4 𝐷 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 V
21prid1 3450 . . . 4 A {A, B}
3 preleq.3 . . . . 5 𝐶 V
43prid1 3450 . . . 4 𝐶 {𝐶, 𝐷}
5 preleq.2 . . . . . 6 B V
6 prexgOLD 3920 . . . . . 6 ((A V B V) → {A, B} V)
71, 5, 6mp2an 404 . . . . 5 {A, B} V
8 preleq.4 . . . . . 6 𝐷 V
9 prexgOLD 3920 . . . . . 6 ((𝐶 V 𝐷 V) → {𝐶, 𝐷} V)
103, 8, 9mp2an 404 . . . . 5 {𝐶, 𝐷} V
111, 7, 3, 10preleq 4217 . . . 4 (((A {A, B} 𝐶 {𝐶, 𝐷}) {A, {A, B}} = {𝐶, {𝐶, 𝐷}}) → (A = 𝐶 {A, B} = {𝐶, 𝐷}))
122, 4, 11mpanl12 414 . . 3 ({A, {A, B}} = {𝐶, {𝐶, 𝐷}} → (A = 𝐶 {A, B} = {𝐶, 𝐷}))
13 preq1 3421 . . . . . 6 (A = 𝐶 → {A, B} = {𝐶, B})
1413eqeq1d 2030 . . . . 5 (A = 𝐶 → ({A, B} = {𝐶, 𝐷} ↔ {𝐶, B} = {𝐶, 𝐷}))
155, 8preqr2 3514 . . . . 5 ({𝐶, B} = {𝐶, 𝐷} → B = 𝐷)
1614, 15syl6bi 152 . . . 4 (A = 𝐶 → ({A, B} = {𝐶, 𝐷} → B = 𝐷))
1716imdistani 422 . . 3 ((A = 𝐶 {A, B} = {𝐶, 𝐷}) → (A = 𝐶 B = 𝐷))
1812, 17syl 14 . 2 ({A, {A, B}} = {𝐶, {𝐶, 𝐷}} → (A = 𝐶 B = 𝐷))
19 preq1 3421 . . . 4 (A = 𝐶 → {A, {A, B}} = {𝐶, {A, B}})
2019adantr 261 . . 3 ((A = 𝐶 B = 𝐷) → {A, {A, B}} = {𝐶, {A, B}})
21 preq12 3423 . . . 4 ((A = 𝐶 B = 𝐷) → {A, B} = {𝐶, 𝐷})
2221preq2d 3428 . . 3 ((A = 𝐶 B = 𝐷) → {𝐶, {A, B}} = {𝐶, {𝐶, 𝐷}})
2320, 22eqtrd 2054 . 2 ((A = 𝐶 B = 𝐷) → {A, {A, B}} = {𝐶, {𝐶, 𝐷}})
2418, 23impbii 117 1 ({A, {A, B}} = {𝐶, {𝐶, 𝐷}} ↔ (A = 𝐶 B = 𝐷))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1228   wcel 1374  Vcvv 2535  {cpr 3351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pr 3918  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-dif 2897  df-un 2899  df-sn 3356  df-pr 3357
This theorem is referenced by: (None)
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