Theorem opeqsn 4070

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 3616	. . 3 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	eqeq1i 2095	. 2 ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶})
5	1	snex 4011	. . 3 ⊢ {𝐴} ∈ V
6		prexg 4029	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
7	1, 2, 6	mp2an 417	. . 3 ⊢ {𝐴, 𝐵} ∈ V
8		opeqsn.3	. . 3 ⊢ 𝐶 ∈ V
9	5, 7, 8	preqsn 3614	. 2 ⊢ ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶))
10		eqcom 2090	. . . . 5 ⊢ ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
11	1, 2, 1	preqsn 3614	. . . . 5 ⊢ ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐴))
12		eqcom 2090	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
13	12	anbi2i 445	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵))
14		anidm 388	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
15	13, 14	bitri 182	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
16	10, 11, 15	3bitri 204	. . . 4 ⊢ ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵)
17	16	anbi1i 446	. . 3 ⊢ (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶))
18		dfsn2 3455	. . . . . . 7 ⊢ {𝐴} = {𝐴, 𝐴}
19		preq2 3515	. . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
20	18, 19	syl5req 2133	. . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
21	20	eqeq1d 2096	. . . . 5 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
22		eqcom 2090	. . . . 5 ⊢ ({𝐴} = 𝐶 ↔ 𝐶 = {𝐴})
23	21, 22	syl6bb 194	. . . 4 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ 𝐶 = {𝐴}))
24	23	pm5.32i 442	. . 3 ⊢ ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))
25	17, 24	bitri 182	. 2 ⊢ (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))
26	4, 9, 25	3bitri 204	1 ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))

Syntax hints:   ∧ wa 102   ↔ wb 103   = wceq 1289   ∈ wcel 1438  Vcvv 2619  {csn 3441  {cpr 3442  ⟨cop 3444

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450

This theorem is referenced by:  relop  4574