Theorem opeqsng 5472

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 4829	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2	1	eqeq1d 2764	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶}))
3		snex 5396	. . . 4 ⊢ {𝐴} ∈ V
4		prex 5395	. . . 4 ⊢ {𝐴, 𝐵} ∈ V
5	3, 4	preqsn 4820	. . 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 2769	. . . . 5 ⊢ ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
8		simpl 486	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
9		simpr 488	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
10	8, 9	preqsnd 4817	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐴 ∧ 𝐵 = 𝐴)))
11		simpr 488	. . . . . . . 8 ⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐴) → 𝐵 = 𝐴)
12		eqid 2762	. . . . . . . . 9 ⊢ 𝐴 = 𝐴
13	12	jctl 531	. . . . . . . 8 ⊢ (𝐵 = 𝐴 → (𝐴 = 𝐴 ∧ 𝐵 = 𝐴))
14	11, 13	impbii 211	. . . . . . 7 ⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐴) ↔ 𝐵 = 𝐴)
15		eqcom 2769	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
16	14, 15	bitri 277	. . . . . 6 ⊢ ((𝐴 = 𝐴 ∧ 𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
17	10, 16	bitrdi 289	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} = {𝐴} ↔ 𝐴 = 𝐵))
18	7, 17	bitrid 285	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵))
19	18	anbi1d 640	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶)))
20		dfsn2 4595	. . . . . . . 8 ⊢ {𝐴} = {𝐴, 𝐴}
21		preq2 4693	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
22	20, 21	eqtr2id 2810	. . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
23	22	eqeq1d 2764	. . . . . 6 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
24		eqcom 2769	. . . . . 6 ⊢ ({𝐴} = 𝐶 ↔ 𝐶 = {𝐴})
25	23, 24	bitrdi 289	. . . . 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 585	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))
28	19, 27	bitrd 281	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))
29	2, 6, 28	3bitrd 307	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {csn 4582  {cpr 4584  ⟨cop 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589

This theorem is referenced by:  opeqsn  5473  snopeqop  5475