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Theorem opthreg 9059
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 9034 (via the preleq 9057 step). See df-op 4550 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) (Proof shortened by AV, 15-Jun-2022.)
Hypotheses
Ref Expression
opthreg.1 𝐴 ∈ V
opthreg.2 𝐵 ∈ V
opthreg.3 𝐶 ∈ V
opthreg.4 𝐷 ∈ V
Assertion
Ref Expression
opthreg ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem opthreg
StepHypRef Expression
1 opthreg.1 . . . . 5 𝐴 ∈ V
21prid1 4674 . . . 4 𝐴 ∈ {𝐴, 𝐵}
3 opthreg.3 . . . . 5 𝐶 ∈ V
43prid1 4674 . . . 4 𝐶 ∈ {𝐶, 𝐷}
5 prex 5309 . . . . 5 {𝐴, 𝐵} ∈ V
65preleq 9057 . . . 4 (((𝐴 ∈ {𝐴, 𝐵} ∧ 𝐶 ∈ {𝐶, 𝐷}) ∧ {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
72, 4, 6mpanl12 700 . . 3 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
8 preq1 4645 . . . . . 6 (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
98eqeq1d 2822 . . . . 5 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
10 opthreg.2 . . . . . 6 𝐵 ∈ V
11 opthreg.4 . . . . . 6 𝐷 ∈ V
1210, 11preqr2 4756 . . . . 5 ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)
139, 12syl6bi 255 . . . 4 (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
1413imdistani 571 . . 3 ((𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))
157, 14syl 17 . 2 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶𝐵 = 𝐷))
16 preq1 4645 . . . 4 (𝐴 = 𝐶 → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
1716adantr 483 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
18 preq12 4647 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
1918preq2d 4652 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐶, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
2017, 19eqtrd 2855 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
2115, 20impbii 211 1 ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1537  wcel 2114  Vcvv 3473  {cpr 4545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-reg 9034
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-eprel 5441  df-fr 5490
This theorem is referenced by: (None)
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