Theorem opeqsn 4169

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 3699	. . 3 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	eqeq1i 2145	. 2 ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶})
5	1	snex 4104	. . 3 ⊢ {𝐴} ∈ V
6		prexg 4128	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
7	1, 2, 6	mp2an 422	. . 3 ⊢ {𝐴, 𝐵} ∈ V
8		opeqsn.3	. . 3 ⊢ 𝐶 ∈ V
9	5, 7, 8	preqsn 3697	. 2 ⊢ ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶))
10		eqcom 2139	. . . . 5 ⊢ ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
11	1, 2, 1	preqsn 3697	. . . . 5 ⊢ ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐴))
12		eqcom 2139	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
13	12	anbi2i 452	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵))
14		anidm 393	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
15	13, 14	bitri 183	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
16	10, 11, 15	3bitri 205	. . . 4 ⊢ ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵)
17	16	anbi1i 453	. . 3 ⊢ (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶))
18		dfsn2 3536	. . . . . . 7 ⊢ {𝐴} = {𝐴, 𝐴}
19		preq2 3596	. . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
20	18, 19	syl5req 2183	. . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
21	20	eqeq1d 2146	. . . . 5 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
22		eqcom 2139	. . . . 5 ⊢ ({𝐴} = 𝐶 ↔ 𝐶 = {𝐴})
23	21, 22	syl6bb 195	. . . 4 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ 𝐶 = {𝐴}))
24	23	pm5.32i 449	. . 3 ⊢ ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))
25	17, 24	bitri 183	. 2 ⊢ (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))
26	4, 9, 25	3bitri 205	1 ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  Vcvv 2681  {csn 3522  {cpr 3523  ⟨cop 3525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531

This theorem is referenced by:  relop  4684