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Theorem opthregOLD 8812
 Description: Obsolete proof of opthreg 8810 as of 15-Jun-2022. Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 8786 (via the preleqOLD 8811 step). See df-op 4405 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
opthreg.1 𝐴 ∈ V
opthreg.2 𝐵 ∈ V
opthreg.3 𝐶 ∈ V
opthreg.4 𝐷 ∈ V
Assertion
Ref Expression
opthregOLD ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem opthregOLD
StepHypRef Expression
1 opthreg.1 . . . . 5 𝐴 ∈ V
21prid1 4529 . . . 4 𝐴 ∈ {𝐴, 𝐵}
3 opthreg.3 . . . . 5 𝐶 ∈ V
43prid1 4529 . . . 4 𝐶 ∈ {𝐶, 𝐷}
5 prex 5141 . . . . 5 {𝐴, 𝐵} ∈ V
6 prex 5141 . . . . 5 {𝐶, 𝐷} ∈ V
71, 5, 3, 6preleqOLD 8811 . . . 4 (((𝐴 ∈ {𝐴, 𝐵} ∧ 𝐶 ∈ {𝐶, 𝐷}) ∧ {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
82, 4, 7mpanl12 692 . . 3 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
9 preq1 4500 . . . . . 6 (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
109eqeq1d 2780 . . . . 5 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
11 opthreg.2 . . . . . 6 𝐵 ∈ V
12 opthreg.4 . . . . . 6 𝐷 ∈ V
1311, 12preqr2 4609 . . . . 5 ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)
1410, 13syl6bi 245 . . . 4 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
1514imdistani 564 . . 3 ((𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))
168, 15syl 17 . 2 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶𝐵 = 𝐷))
17 preq1 4500 . . . 4 (𝐴 = 𝐶 → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
1817adantr 474 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
19 preq12 4502 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
2019preq2d 4507 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐶, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
2118, 20eqtrd 2814 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
2216, 21impbii 201 1 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398  {cpr 4400 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-reg 8786 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-eprel 5266  df-fr 5314 This theorem is referenced by: (None)
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