Theorem opeqsng 5487

Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.)

Assertion

Ref	Expression
opeqsng	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))

Proof of Theorem opeqsng

Step	Hyp	Ref	Expression
1		dfopg 4840	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2	1	eqeq1d 2771	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶}))
3		snex 5411	. . . 4 ⊢ {𝐴} ∈ V
4		prex 5410	. . . 4 ⊢ {𝐴, 𝐵} ∈ V
5	3, 4	preqsn 4831	. . 3 ⊢ ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶))
6	5	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶)))
7		eqcom 2776	. . . . 5 ⊢ ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
8		simpl 487	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
9		simpr 489	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
10	8, 9	preqsnd 4828	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐴 ∧ 𝐵 = 𝐴)))
11		simpr 489	. . . . . . . 8 ⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐴) → 𝐵 = 𝐴)
12		eqid 2769	. . . . . . . . 9 ⊢ 𝐴 = 𝐴
13	12	jctl 532	. . . . . . . 8 ⊢ (𝐵 = 𝐴 → (𝐴 = 𝐴 ∧ 𝐵 = 𝐴))
14	11, 13	impbii 212	. . . . . . 7 ⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐴) ↔ 𝐵 = 𝐴)
15		eqcom 2776	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
16	14, 15	bitri 278	. . . . . 6 ⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
17	10, 16	bitrdi 290	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} = {𝐴} ↔ 𝐴 = 𝐵))
18	7, 17	bitrid 286	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵))
19	18	anbi1d 642	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶)))
20		dfsn2 4607	. . . . . . . 8 ⊢ {𝐴} = {𝐴, 𝐴}
21		preq2 4705	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
22	20, 21	eqtr2id 2817	. . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
23	22	eqeq1d 2771	. . . . . 6 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
24		eqcom 2776	. . . . . 6 ⊢ ({𝐴} = 𝐶 ↔ 𝐶 = {𝐴})
25	23, 24	bitrdi 290	. . . . 5 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ 𝐶 = {𝐴}))
26	25	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ 𝐶 = {𝐴})))
27	26	pm5.32d 587	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))
28	19, 27	bitrd 282	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))
29	2, 6, 28	3bitrd 308	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {csn 4594  {cpr 4596  ⟨cop 4600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601

This theorem is referenced by:  opeqsn  5488  snopeqop  5490