Theorem opeqsn 3963

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 3522	. . 3 ⊢ ⟨A, B⟩ = {{A}, {A, B}}
4	3	eqeq1i 2029	. 2 ⊢ (⟨A, B⟩ = {𝐶} ↔ {{A}, {A, B}} = {𝐶})
5		snexgOLD 3909	. . . 4 ⊢ (A ∈ V → {A} ∈ V)
6	1, 5	ax-mp 7	. . 3 ⊢ {A} ∈ V
7		prexgOLD 3920	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → {A, B} ∈ V)
8	1, 2, 7	mp2an 404	. . 3 ⊢ {A, B} ∈ V
9		opeqsn.3	. . 3 ⊢ 𝐶 ∈ V
10	6, 8, 9	preqsn 3520	. 2 ⊢ ({{A}, {A, B}} = {𝐶} ↔ ({A} = {A, B} ∧ {A, B} = 𝐶))
11		eqcom 2024	. . . . 5 ⊢ ({A} = {A, B} ↔ {A, B} = {A})
12	1, 2, 1	preqsn 3520	. . . . 5 ⊢ ({A, B} = {A} ↔ (A = B ∧ B = A))
13		eqcom 2024	. . . . . . 7 ⊢ (B = A ↔ A = B)
14	13	anbi2i 433	. . . . . 6 ⊢ ((A = B ∧ B = A) ↔ (A = B ∧ A = B))
15		anidm 376	. . . . . 6 ⊢ ((A = B ∧ A = B) ↔ A = B)
16	14, 15	bitri 173	. . . . 5 ⊢ ((A = B ∧ B = A) ↔ A = B)
17	11, 12, 16	3bitri 195	. . . 4 ⊢ ({A} = {A, B} ↔ A = B)
18	17	anbi1i 434	. . 3 ⊢ (({A} = {A, B} ∧ {A, B} = 𝐶) ↔ (A = B ∧ {A, B} = 𝐶))
19		dfsn2 3364	. . . . . . 7 ⊢ {A} = {A, A}
20		preq2 3422	. . . . . . 7 ⊢ (A = B → {A, A} = {A, B})
21	19, 20	syl5req 2067	. . . . . 6 ⊢ (A = B → {A, B} = {A})
22	21	eqeq1d 2030	. . . . 5 ⊢ (A = B → ({A, B} = 𝐶 ↔ {A} = 𝐶))
23		eqcom 2024	. . . . 5 ⊢ ({A} = 𝐶 ↔ 𝐶 = {A})
24	22, 23	syl6bb 185	. . . 4 ⊢ (A = B → ({A, B} = 𝐶 ↔ 𝐶 = {A}))
25	24	pm5.32i 430	. . 3 ⊢ ((A = B ∧ {A, B} = 𝐶) ↔ (A = B ∧ 𝐶 = {A}))
26	18, 25	bitri 173	. 2 ⊢ (({A} = {A, B} ∧ {A, B} = 𝐶) ↔ (A = B ∧ 𝐶 = {A}))
27	4, 10, 26	3bitri 195	1 ⊢ (⟨A, B⟩ = {𝐶} ↔ (A = B ∧ 𝐶 = {A}))

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374  Vcvv 2535  {csn 3350  {cpr 3351  ⟨cop 3353

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359

This theorem is referenced by:  relop  4413