Theorem opeqsn 4112

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 3651	. . 3 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	eqeq1i 2107	. 2 ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶})
5	1	snex 4049	. . 3 ⊢ {𝐴} ∈ V
6		prexg 4071	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
7	1, 2, 6	mp2an 420	. . 3 ⊢ {𝐴, 𝐵} ∈ V
8		opeqsn.3	. . 3 ⊢ 𝐶 ∈ V
9	5, 7, 8	preqsn 3649	. 2 ⊢ ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶))
10		eqcom 2102	. . . . 5 ⊢ ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
11	1, 2, 1	preqsn 3649	. . . . 5 ⊢ ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐴))
12		eqcom 2102	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
13	12	anbi2i 448	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵))
14		anidm 391	. . . . . 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 449	. . 3 ⊢ (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶))
18		dfsn2 3488	. . . . . . 7 ⊢ {𝐴} = {𝐴, 𝐴}
19		preq2 3548	. . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
20	18, 19	syl5req 2145	. . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
21	20	eqeq1d 2108	. . . . 5 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
22		eqcom 2102	. . . . 5 ⊢ ({𝐴} = 𝐶 ↔ 𝐶 = {𝐴})
23	21, 22	syl6bb 195	. . . 4 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ 𝐶 = {𝐴}))
24	23	pm5.32i 445	. . 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 1299   ∈ wcel 1448  Vcvv 2641  {csn 3474  {cpr 3475  ⟨cop 3477

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483

This theorem is referenced by:  relop  4627