ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opthreg Structured version   GIF version

Theorem opthreg 4234
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4220 (via the preleq 4233 step). See df-op 3376 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 3467 . . . 4 A {A, B}
3 preleq.3 . . . . 5 𝐶 V
43prid1 3467 . . . 4 𝐶 {𝐶, 𝐷}
5 preleq.2 . . . . . 6 B V
6 prexgOLD 3937 . . . . . 6 ((A V B V) → {A, B} V)
71, 5, 6mp2an 402 . . . . 5 {A, B} V
8 preleq.4 . . . . . 6 𝐷 V
9 prexgOLD 3937 . . . . . 6 ((𝐶 V 𝐷 V) → {𝐶, 𝐷} V)
103, 8, 9mp2an 402 . . . . 5 {𝐶, 𝐷} V
111, 7, 3, 10preleq 4233 . . . 4 (((A {A, B} 𝐶 {𝐶, 𝐷}) {A, {A, B}} = {𝐶, {𝐶, 𝐷}}) → (A = 𝐶 {A, B} = {𝐶, 𝐷}))
122, 4, 11mpanl12 412 . . 3 ({A, {A, B}} = {𝐶, {𝐶, 𝐷}} → (A = 𝐶 {A, B} = {𝐶, 𝐷}))
13 preq1 3438 . . . . . 6 (A = 𝐶 → {A, B} = {𝐶, B})
1413eqeq1d 2045 . . . . 5 (A = 𝐶 → ({A, B} = {𝐶, 𝐷} ↔ {𝐶, B} = {𝐶, 𝐷}))
155, 8preqr2 3531 . . . . 5 ({𝐶, B} = {𝐶, 𝐷} → B = 𝐷)
1614, 15syl6bi 152 . . . 4 (A = 𝐶 → ({A, B} = {𝐶, 𝐷} → B = 𝐷))
1716imdistani 419 . . 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 3438 . . . 4 (A = 𝐶 → {A, {A, B}} = {𝐶, {A, B}})
2019adantr 261 . . 3 ((A = 𝐶 B = 𝐷) → {A, {A, B}} = {𝐶, {A, B}})
21 preq12 3440 . . . 4 ((A = 𝐶 B = 𝐷) → {A, B} = {𝐶, 𝐷})
2221preq2d 3445 . . 3 ((A = 𝐶 B = 𝐷) → {𝐶, {A, B}} = {𝐶, {𝐶, 𝐷}})
2320, 22eqtrd 2069 . 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 1242   wcel 1390  Vcvv 2551  {cpr 3368
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pr 3935  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator