Theorem opeqsng 5478

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 2765	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶}))
3		snex 5400	. . . 4 ⊢ {𝐴} ∈ V
4		prex 5399	. . . 4 ⊢ {𝐴, 𝐵} ∈ V
5	3, 4	preqsn 4819	. . 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 2770	. . . . 5 ⊢ ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
8		simpl 488	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
9		simpr 490	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
10	8, 9	preqsnd 4816	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐴 ∧ 𝐵 = 𝐴)))
11		simpr 490	. . . . . . . 8 ⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐴) → 𝐵 = 𝐴)
12		eqid 2763	. . . . . . . . 9 ⊢ 𝐴 = 𝐴
13	12	jctl 533	. . . . . . . 8 ⊢ (𝐵 = 𝐴 → (𝐴 = 𝐴 ∧ 𝐵 = 𝐴))
14	11, 13	impbii 212	. . . . . . 7 ⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐴) ↔ 𝐵 = 𝐴)
15		eqcom 2770	. . . . . . 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 643	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶)))
20		dfsn2 4596	. . . . . . . 8 ⊢ {𝐴} = {𝐴, 𝐴}
21		preq2 4694	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
22	20, 21	eqtr2id 2811	. . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
23	22	eqeq1d 2765	. . . . . 6 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
24		eqcom 2770	. . . . . 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 588	. . 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})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  {csn 4587  {cpr 4589  ⟨cop 4837

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3455  df-ss 3900  df-dif 4049  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838

This theorem is used by:  opeqsn  5479  snopeqop  5481