Theorem opksnelsik 4265

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 4111	. . 3 ⊢ {A} ∈ V
2		snex 4111	. . 3 ⊢ {B} ∈ V
3		opkelsikg 4264	. . 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 2862	. . . . . . 7 ⊢ x ∈ V
7	6	sneqb 3876	. . . . . 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 2862	. . . . . . 7 ⊢ y ∈ V
11	10	sneqb 3876	. . . . . 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 4059	. . . . 5 ⊢ (x = A → ⟪x, y⟫ = ⟪A, y⟫)
19	18	eleq1d 2419	. . . 4 ⊢ (x = A → (⟪x, y⟫ ∈ C ↔ ⟪A, y⟫ ∈ C))
20		opkeq2 4060	. . . . 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 2896	. . 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 2859  {csn 3737  ⟪copk 4057   SI_k csik 4181

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 4078  ax-sn 4087

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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058  df-sik 4192

This theorem is referenced by:  opkelimagekg  4271  sikexg  4296  dfimak2  4298  dfaddc2  4381  dfnnc2  4395  nnsucelrlem1  4424  ltfinex  4464  eqpwrelk  4478  eqpw1relk  4479  ncfinraiselem2  4480  ncfinlowerlem1  4482  eqtfinrelk  4486  evenfinex  4503  oddfinex  4504  evenodddisjlem1  4515  nnadjoinlem1  4519  nnpweqlem1  4522  srelk  4524  sfintfinlem1  4531  tfinnnlem1  4533  spfinex  4537  setconslem2  4732  setconslem3  4733  setconslem7  4737  dfswap2  4741