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Theorem opthreg 4362
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4343 (via the preleq 4361 step). See df-op 3450 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 = 𝐶𝐵 = 𝐷))

Proof of Theorem opthreg
StepHypRef Expression
1 preleq.1 . . . . 5 𝐴 ∈ V
21prid1 3543 . . . 4 𝐴 ∈ {𝐴, 𝐵}
3 preleq.3 . . . . 5 𝐶 ∈ V
43prid1 3543 . . . 4 𝐶 ∈ {𝐶, 𝐷}
5 preleq.2 . . . . . 6 𝐵 ∈ V
6 prexg 4029 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
71, 5, 6mp2an 417 . . . . 5 {𝐴, 𝐵} ∈ V
8 preleq.4 . . . . . 6 𝐷 ∈ V
9 prexg 4029 . . . . . 6 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → {𝐶, 𝐷} ∈ V)
103, 8, 9mp2an 417 . . . . 5 {𝐶, 𝐷} ∈ V
111, 7, 3, 10preleq 4361 . . . 4 (((𝐴 ∈ {𝐴, 𝐵} ∧ 𝐶 ∈ {𝐶, 𝐷}) ∧ {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
122, 4, 11mpanl12 427 . . 3 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
13 preq1 3514 . . . . . 6 (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
1413eqeq1d 2096 . . . . 5 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
155, 8preqr2 3608 . . . . 5 ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)
1614, 15syl6bi 161 . . . 4 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
1716imdistani 434 . . 3 ((𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))
1812, 17syl 14 . 2 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶𝐵 = 𝐷))
19 preq1 3514 . . . 4 (𝐴 = 𝐶 → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
2019adantr 270 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
21 preq12 3516 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
2221preq2d 3521 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐶, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
2320, 22eqtrd 2120 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
2418, 23impbii 124 1 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1289  wcel 1438  Vcvv 2619  {cpr 3442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pr 4027  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-dif 2999  df-un 3001  df-sn 3447  df-pr 3448
This theorem is referenced by: (None)
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