Theorem opthreg 4540

Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4521 (via the preleq 4539 step). See df-op 3592 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 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem opthreg

Step	Hyp	Ref	Expression
1		preleq.1	. . . . 5 ⊢ 𝐴 ∈ V
2	1	prid1 3689	. . . 4 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3		preleq.3	. . . . 5 ⊢ 𝐶 ∈ V
4	3	prid1 3689	. . . 4 ⊢ 𝐶 ∈ {𝐶, 𝐷}
5		preleq.2	. . . . . 6 ⊢ 𝐵 ∈ V
6		prexg 4196	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
7	1, 5, 6	mp2an 424	. . . . 5 ⊢ {𝐴, 𝐵} ∈ V
8		preleq.4	. . . . . 6 ⊢ 𝐷 ∈ V
9		prexg 4196	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → {𝐶, 𝐷} ∈ V)
10	3, 8, 9	mp2an 424	. . . . 5 ⊢ {𝐶, 𝐷} ∈ V
11	1, 7, 3, 10	preleq 4539	. . . 4 ⊢ (((𝐴 ∈ {𝐴, 𝐵} ∧ 𝐶 ∈ {𝐶, 𝐷}) ∧ {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
12	2, 4, 11	mpanl12 434	. . 3 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}))
13		preq1 3660	. . . . . 6 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
14	13	eqeq1d 2179	. . . . 5 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
15	5, 8	preqr2 3756	. . . . 5 ⊢ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)
16	14, 15	syl6bi 162	. . . 4 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
17	16	imdistani 443	. . 3 ⊢ ((𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
18	12, 17	syl 14	. 2 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
19		preq1 3660	. . . 4 ⊢ (𝐴 = 𝐶 → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
20	19	adantr 274	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}})
21		preq12 3662	. . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
22	21	preq2d 3667	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐶, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
23	20, 22	eqtrd 2203	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}})
24	18, 23	impbii 125	1 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  Vcvv 2730  {cpr 3584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pr 4194  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-un 3125  df-sn 3589  df-pr 3590

This theorem is referenced by: (None)