Theorem opeqsn 4249

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 3775	. . 3 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	eqeq1i 2185	. 2 ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶})
5	1	snex 4182	. . 3 ⊢ {𝐴} ∈ V
6		prexg 4208	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
7	1, 2, 6	mp2an 426	. . 3 ⊢ {𝐴, 𝐵} ∈ V
8		opeqsn.3	. . 3 ⊢ 𝐶 ∈ V
9	5, 7, 8	preqsn 3773	. 2 ⊢ ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶))
10		eqcom 2179	. . . . 5 ⊢ ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
11	1, 2, 1	preqsn 3773	. . . . 5 ⊢ ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐴))
12		eqcom 2179	. . . . . . 7 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
13	12	anbi2i 457	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵))
14		anidm 396	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
15	13, 14	bitri 184	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
16	10, 11, 15	3bitri 206	. . . 4 ⊢ ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵)
17	16	anbi1i 458	. . 3 ⊢ (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶))
18		dfsn2 3605	. . . . . . 7 ⊢ {𝐴} = {𝐴, 𝐴}
19		preq2 3669	. . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
20	18, 19	eqtr2id 2223	. . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
21	20	eqeq1d 2186	. . . . 5 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
22		eqcom 2179	. . . . 5 ⊢ ({𝐴} = 𝐶 ↔ 𝐶 = {𝐴})
23	21, 22	bitrdi 196	. . . 4 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ 𝐶 = {𝐴}))
24	23	pm5.32i 454	. . 3 ⊢ ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))
25	17, 24	bitri 184	. 2 ⊢ (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))
26	4, 9, 25	3bitri 206	1 ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  Vcvv 2737  {csn 3591  {cpr 3592  ⟨cop 3594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600

This theorem is referenced by:  relop  4773