Theorem opeqsng 5411

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 4799	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2	1	eqeq1d 2740	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶}))
3		snex 5349	. . . 4 ⊢ {𝐴} ∈ V
4		prex 5350	. . . 4 ⊢ {𝐴, 𝐵} ∈ V
5	3, 4	preqsn 4789	. . 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 2745	. . . . 5 ⊢ ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
8		simpl 482	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
9		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
10	8, 9	preqsnd 4786	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐴 ∧ 𝐵 = 𝐴)))
11		simpr 484	. . . . . . . 8 ⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐴) → 𝐵 = 𝐴)
12		eqid 2738	. . . . . . . . 9 ⊢ 𝐴 = 𝐴
13	12	jctl 523	. . . . . . . 8 ⊢ (𝐵 = 𝐴 → (𝐴 = 𝐴 ∧ 𝐵 = 𝐴))
14	11, 13	impbii 208	. . . . . . 7 ⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐴) ↔ 𝐵 = 𝐴)
15		eqcom 2745	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
16	14, 15	bitri 274	. . . . . 6 ⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
17	10, 16	bitrdi 286	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} = {𝐴} ↔ 𝐴 = 𝐵))
18	7, 17	syl5bb 282	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵))
19	18	anbi1d 629	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶)))
20		dfsn2 4571	. . . . . . . 8 ⊢ {𝐴} = {𝐴, 𝐴}
21		preq2 4667	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
22	20, 21	eqtr2id 2792	. . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
23	22	eqeq1d 2740	. . . . . 6 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
24		eqcom 2745	. . . . . 6 ⊢ ({𝐴} = 𝐶 ↔ 𝐶 = {𝐴})
25	23, 24	bitrdi 286	. . . . 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 576	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))
28	19, 27	bitrd 278	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))
29	2, 6, 28	3bitrd 304	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {csn 4558  {cpr 4560  ⟨cop 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565

This theorem is referenced by:  opeqsn  5412  snopeqop  5414