Theorem opksnelsik 4266

Description: Membership of an ordered pair of singletons in a Kuratowski singleton image. (Contributed by SF, 13-Jan-2015.)

Hypotheses

Ref	Expression
opksnelsik.1	⊢ A ∈ V
opksnelsik.2	⊢ B ∈ V

Assertion

Ref	Expression
opksnelsik	⊢ (⟪{A}, {B}⟫ ∈ SIk C ↔ ⟪A, B⟫ ∈ C)

Proof of Theorem opksnelsik

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4112	. . 3 ⊢ {A} ∈ V
2		snex 4112	. . 3 ⊢ {B} ∈ V
3		opkelsikg 4265	. . 3 ⊢ (({A} ∈ V ∧ {B} ∈ V) → (⟪{A}, {B}⟫ ∈ SIk C ↔ ∃x∃y({A} = {x} ∧ {B} = {y} ∧ ⟪x, y⟫ ∈ C)))
4	1, 2, 3	mp2an 653	. 2 ⊢ (⟪{A}, {B}⟫ ∈ SIk C ↔ ∃x∃y({A} = {x} ∧ {B} = {y} ∧ ⟪x, y⟫ ∈ C))
5		eqcom 2355	. . . . . 6 ⊢ ({A} = {x} ↔ {x} = {A})
6		vex 2863	. . . . . . 7 ⊢ x ∈ V
7	6	sneqb 3877	. . . . . 6 ⊢ ({x} = {A} ↔ x = A)
8	5, 7	bitri 240	. . . . 5 ⊢ ({A} = {x} ↔ x = A)
9		eqcom 2355	. . . . . 6 ⊢ ({B} = {y} ↔ {y} = {B})
10		vex 2863	. . . . . . 7 ⊢ y ∈ V
11	10	sneqb 3877	. . . . . 6 ⊢ ({y} = {B} ↔ y = B)
12	9, 11	bitri 240	. . . . 5 ⊢ ({B} = {y} ↔ y = B)
13		biid 227	. . . . 5 ⊢ (⟪x, y⟫ ∈ C ↔ ⟪x, y⟫ ∈ C)
14	8, 12, 13	3anbi123i 1140	. . . 4 ⊢ (({A} = {x} ∧ {B} = {y} ∧ ⟪x, y⟫ ∈ C) ↔ (x = A ∧ y = B ∧ ⟪x, y⟫ ∈ C))
15	14	2exbii 1583	. . 3 ⊢ (∃x∃y({A} = {x} ∧ {B} = {y} ∧ ⟪x, y⟫ ∈ C) ↔ ∃x∃y(x = A ∧ y = B ∧ ⟪x, y⟫ ∈ C))
16		opksnelsik.1	. . . 4 ⊢ A ∈ V
17		opksnelsik.2	. . . 4 ⊢ B ∈ V
18		opkeq1 4060	. . . . 5 ⊢ (x = A → ⟪x, y⟫ = ⟪A, y⟫)
19	18	eleq1d 2419	. . . 4 ⊢ (x = A → (⟪x, y⟫ ∈ C ↔ ⟪A, y⟫ ∈ C))
20		opkeq2 4061	. . . . 5 ⊢ (y = B → ⟪A, y⟫ = ⟪A, B⟫)
21	20	eleq1d 2419	. . . 4 ⊢ (y = B → (⟪A, y⟫ ∈ C ↔ ⟪A, B⟫ ∈ C))
22	16, 17, 19, 21	ceqsex2v 2897	. . 3 ⊢ (∃x∃y(x = A ∧ y = B ∧ ⟪x, y⟫ ∈ C) ↔ ⟪A, B⟫ ∈ C)
23	15, 22	bitri 240	. 2 ⊢ (∃x∃y({A} = {x} ∧ {B} = {y} ∧ ⟪x, y⟫ ∈ C) ↔ ⟪A, B⟫ ∈ C)
24	4, 23	bitri 240	1 ⊢ (⟪{A}, {B}⟫ ∈ SIk C ↔ ⟪A, B⟫ ∈ C)

Syntax hints:   ↔ wb 176   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {csn 3738  ⟪copk 4058   SI_k csik 4182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059  df-sik 4193

This theorem is referenced by:  opkelimagekg  4272  sikexg  4297  dfimak2  4299  dfaddc2  4382  dfnnc2  4396  nnsucelrlem1  4425  ltfinex  4465  eqpwrelk  4479  eqpw1relk  4480  ncfinraiselem2  4481  ncfinlowerlem1  4483  eqtfinrelk  4487  evenfinex  4504  oddfinex  4505  evenodddisjlem1  4516  nnadjoinlem1  4520  nnpweqlem1  4523  srelk  4525  sfintfinlem1  4532  tfinnnlem1  4534  spfinex  4538  setconslem2  4733  setconslem3  4734  setconslem7  4738  dfswap2  4742