Theorem opeqsn 4892

Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Hypotheses

Ref	Expression
opeqsn.1	⊢ 𝐴 ∈ V
opeqsn.2	⊢ 𝐵 ∈ V
opeqsn.3	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
opeqsn	⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))

Proof of Theorem opeqsn

Step	Hyp	Ref	Expression
1		opeqsn.1	. . . 4 ⊢ 𝐴 ∈ V
2		opeqsn.2	. . . 4 ⊢ 𝐵 ∈ V
3	1, 2	dfop 4339	. . 3 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	eqeq1i 2615	. 2 ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶})
5		snex 4835	. . 3 ⊢ {𝐴} ∈ V
6		prex 4836	. . 3 ⊢ {𝐴, 𝐵} ∈ V
7		opeqsn.3	. . 3 ⊢ 𝐶 ∈ V
8	5, 6, 7	preqsn 4331	. 2 ⊢ ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶))
9		eqcom 2617	. . . . 5 ⊢ ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
10	1, 2, 1	preqsn 4331	. . . . 5 ⊢ ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐴))
11		eqcom 2617	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
12	11	anbi2i 726	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵))
13		anidm 674	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
14	12, 13	bitri 263	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
15	9, 10, 14	3bitri 285	. . . 4 ⊢ ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵)
16	15	anbi1i 727	. . 3 ⊢ (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶))
17		dfsn2 4138	. . . . . . 7 ⊢ {𝐴} = {𝐴, 𝐴}
18		preq2 4213	. . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
19	17, 18	syl5req 2657	. . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
20	19	eqeq1d 2612	. . . . 5 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
21		eqcom 2617	. . . . 5 ⊢ ({𝐴} = 𝐶 ↔ 𝐶 = {𝐴})
22	20, 21	syl6bb 275	. . . 4 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ 𝐶 = {𝐴}))
23	22	pm5.32i 667	. . 3 ⊢ ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))
24	16, 23	bitri 263	. 2 ⊢ (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))
25	4, 8, 24	3bitri 285	1 ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  {cpr 4127  ⟨cop 4131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132

This theorem is referenced by:  snopeqop  4894  propeqop  4895  relop  5194