Theorem opthreg 9075

Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 9050 (via the preleq 9073 step). See df-op 4567 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 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem opthreg

Step	Hyp	Ref	Expression
1		opthreg.1	. . . . 5 ⊢ 𝐴 ∈ V
2	1	prid1 4691	. . . 4 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3		opthreg.3	. . . . 5 ⊢ 𝐶 ∈ V
4	3	prid1 4691	. . . 4 ⊢ 𝐶 ∈ {𝐶, 𝐷}
5		prex 5324	. . . . 5 ⊢ {𝐴, 𝐵} ∈ V
6	5	preleq 9073	. . . 4 ⊢ (((𝐴 ∈ {𝐴, 𝐵} ∧ 𝐶 ∈ {𝐶, 𝐷}) ∧ {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
7	2, 4, 6	mpanl12 700	. . 3 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
8		preq1 4662	. . . . . 6 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
9	8	eqeq1d 2823	. . . . 5 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
10		opthreg.2	. . . . . 6 ⊢ 𝐵 ∈ V
11		opthreg.4	. . . . . 6 ⊢ 𝐷 ∈ V
12	10, 11	preqr2 4773	. . . . 5 ⊢ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)
13	9, 12	syl6bi 255	. . . 4 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
14	13	imdistani 571	. . 3 ⊢ ((𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
15	7, 14	syl 17	. 2 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
16		preq1 4662	. . . 4 ⊢ (𝐴 = 𝐶 → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
17	16	adantr 483	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
18		preq12 4664	. . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
19	18	preq2d 4669	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐶, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
20	17, 19	eqtrd 2856	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
21	15, 20	impbii 211	1 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494  {cpr 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-reg 9050

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-eprel 5459  df-fr 5508

This theorem is referenced by: (None)