Theorem sikss1c1c 4268

Description: A Kuratowski singleton image is a subset of (1_c ×_k 1_c). (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
sikss1c1c	⊢ SIk A ⊆ (1c ×k 1c)

Proof of Theorem sikss1c1c

Dummy variables x y z w t a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sik 4193	. . . . 5 ⊢ SIk A = {t ∣ ∃z∃w(t = ⟪z, w⟫ ∧ ∃a∃b(z = {a} ∧ w = {b} ∧ ⟪a, b⟫ ∈ A))}
2		eqeq1 2359	. . . . . . 7 ⊢ (z = x → (z = {a} ↔ x = {a}))
3	2	3anbi1d 1256	. . . . . 6 ⊢ (z = x → ((z = {a} ∧ w = {b} ∧ ⟪a, b⟫ ∈ A) ↔ (x = {a} ∧ w = {b} ∧ ⟪a, b⟫ ∈ A)))
4	3	2exbidv 1628	. . . . 5 ⊢ (z = x → (∃a∃b(z = {a} ∧ w = {b} ∧ ⟪a, b⟫ ∈ A) ↔ ∃a∃b(x = {a} ∧ w = {b} ∧ ⟪a, b⟫ ∈ A)))
5		eqeq1 2359	. . . . . . 7 ⊢ (w = y → (w = {b} ↔ y = {b}))
6	5	3anbi2d 1257	. . . . . 6 ⊢ (w = y → ((x = {a} ∧ w = {b} ∧ ⟪a, b⟫ ∈ A) ↔ (x = {a} ∧ y = {b} ∧ ⟪a, b⟫ ∈ A)))
7	6	2exbidv 1628	. . . . 5 ⊢ (w = y → (∃a∃b(x = {a} ∧ w = {b} ∧ ⟪a, b⟫ ∈ A) ↔ ∃a∃b(x = {a} ∧ y = {b} ∧ ⟪a, b⟫ ∈ A)))
8		vex 2863	. . . . 5 ⊢ x ∈ V
9		vex 2863	. . . . 5 ⊢ y ∈ V
10	1, 4, 7, 8, 9	opkelopkab 4247	. . . 4 ⊢ (⟪x, y⟫ ∈ SIk A ↔ ∃a∃b(x = {a} ∧ y = {b} ∧ ⟪a, b⟫ ∈ A))
11		opkeq12 4062	. . . . . . 7 ⊢ ((x = {a} ∧ y = {b}) → ⟪x, y⟫ = ⟪{a}, {b}⟫)
12		vex 2863	. . . . . . . . 9 ⊢ a ∈ V
13	12	snel1c 4141	. . . . . . . 8 ⊢ {a} ∈ 1c
14		vex 2863	. . . . . . . . 9 ⊢ b ∈ V
15	14	snel1c 4141	. . . . . . . 8 ⊢ {b} ∈ 1c
16		opkelxpkg 4248	. . . . . . . . 9 ⊢ (({a} ∈ 1c ∧ {b} ∈ 1c) → (⟪{a}, {b}⟫ ∈ (1c ×k 1c) ↔ ({a} ∈ 1c ∧ {b} ∈ 1c)))
17	13, 15, 16	mp2an 653	. . . . . . . 8 ⊢ (⟪{a}, {b}⟫ ∈ (1c ×k 1c) ↔ ({a} ∈ 1c ∧ {b} ∈ 1c))
18	13, 15, 17	mpbir2an 886	. . . . . . 7 ⊢ ⟪{a}, {b}⟫ ∈ (1c ×k 1c)
19	11, 18	syl6eqel 2441	. . . . . 6 ⊢ ((x = {a} ∧ y = {b}) → ⟪x, y⟫ ∈ (1c ×k 1c))
20	19	3adant3 975	. . . . 5 ⊢ ((x = {a} ∧ y = {b} ∧ ⟪a, b⟫ ∈ A) → ⟪x, y⟫ ∈ (1c ×k 1c))
21	20	exlimivv 1635	. . . 4 ⊢ (∃a∃b(x = {a} ∧ y = {b} ∧ ⟪a, b⟫ ∈ A) → ⟪x, y⟫ ∈ (1c ×k 1c))
22	10, 21	sylbi 187	. . 3 ⊢ (⟪x, y⟫ ∈ SIk A → ⟪x, y⟫ ∈ (1c ×k 1c))
23	22	gen2 1547	. 2 ⊢ ∀x∀y(⟪x, y⟫ ∈ SIk A → ⟪x, y⟫ ∈ (1c ×k 1c))
24		sikssvvk 4267	. . 3 ⊢ SIk A ⊆ (V ×k V)
25		ssrelk 4212	. . 3 ⊢ ( SIk A ⊆ (V ×k V) → ( SIk A ⊆ (1c ×k 1c) ↔ ∀x∀y(⟪x, y⟫ ∈ SIk A → ⟪x, y⟫ ∈ (1c ×k 1c))))
26	24, 25	ax-mp 5	. 2 ⊢ ( SIk A ⊆ (1c ×k 1c) ↔ ∀x∀y(⟪x, y⟫ ∈ SIk A → ⟪x, y⟫ ∈ (1c ×k 1c)))
27	23, 26	mpbir 200	1 ⊢ SIk A ⊆ (1c ×k 1c)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ⊆ wss 3258  {csn 3738  ⟪copk 4058  1_cc1c 4135   ×_k cxpk 4175   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-ral 2620  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-1c 4137  df-xpk 4186  df-sik 4193

This theorem is referenced by:  opkelimagekg  4272  sikexg  4297  dfnnc2  4396