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Theorem opthregOLD 8678
Description: Obsolete proof of opthreg 8676 as of 15-Jun-2022. Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 8652 (via the preleqOLD 8677 step). See df-op 4321 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 4431 . . . 4 𝐴 ∈ {𝐴, 𝐵}
3 opthreg.3 . . . . 5 𝐶 ∈ V
43prid1 4431 . . . 4 𝐶 ∈ {𝐶, 𝐷}
5 prex 5037 . . . . 5 {𝐴, 𝐵} ∈ V
6 prex 5037 . . . . 5 {𝐶, 𝐷} ∈ V
71, 5, 3, 6preleqOLD 8677 . . . 4 (((𝐴 ∈ {𝐴, 𝐵} ∧ 𝐶 ∈ {𝐶, 𝐷}) ∧ {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
82, 4, 7mpanl12 674 . . 3 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
9 preq1 4402 . . . . . 6 (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
109eqeq1d 2772 . . . . 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 4510 . . . . 5 ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)
1410, 13syl6bi 243 . . . 4 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
1514imdistani 550 . . 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 4402 . . . 4 (𝐴 = 𝐶 → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
1817adantr 466 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
19 preq12 4404 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
2019preq2d 4409 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐶, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
2118, 20eqtrd 2804 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
2216, 21impbii 199 1 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  {cpr 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-reg 8652
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-eprel 5162  df-fr 5208
This theorem is referenced by: (None)
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