Theorem ndisjrelk 4324

Description: Membership in a particular Kuratowski relationship is equivalent to non-disjointedness. (Contributed by SF, 15-Jan-2015.)

Hypotheses

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

Assertion

Ref	Expression
ndisjrelk	⊢ (⟪A, B⟫ ∈ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ↔ (A ∩ B) ≠ ∅)

Proof of Theorem ndisjrelk

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

Step	Hyp	Ref	Expression
1		snex 4112	. . . . 5 ⊢ {{{x}}} ∈ V
2		opkeq1 4060	. . . . . 6 ⊢ (t = {{{x}}} → ⟪t, ⟪A, B⟫⟫ = ⟪{{{x}}}, ⟪A, B⟫⟫)
3	2	eleq1d 2419	. . . . 5 ⊢ (t = {{{x}}} → (⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk ) ↔ ⟪{{{x}}}, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )))
4	1, 3	ceqsexv 2895	. . . 4 ⊢ (∃t(t = {{{x}}} ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )) ↔ ⟪{{{x}}}, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk ))
5		elin 3220	. . . . 5 ⊢ (⟪{{{x}}}, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk ) ↔ (⟪{{{x}}}, ⟪A, B⟫⟫ ∈ Ins3k Sk ∧ ⟪{{{x}}}, ⟪A, B⟫⟫ ∈ Ins2k Sk ))
6		snex 4112	. . . . . . . 8 ⊢ {x} ∈ V
7		ndisjrelk.1	. . . . . . . 8 ⊢ A ∈ V
8		ndisjrelk.2	. . . . . . . 8 ⊢ B ∈ V
9	6, 7, 8	otkelins3k 4257	. . . . . . 7 ⊢ (⟪{{{x}}}, ⟪A, B⟫⟫ ∈ Ins3k Sk ↔ ⟪{x}, A⟫ ∈ Sk )
10		vex 2863	. . . . . . . 8 ⊢ x ∈ V
11	10, 7	elssetk 4271	. . . . . . 7 ⊢ (⟪{x}, A⟫ ∈ Sk ↔ x ∈ A)
12	9, 11	bitri 240	. . . . . 6 ⊢ (⟪{{{x}}}, ⟪A, B⟫⟫ ∈ Ins3k Sk ↔ x ∈ A)
13	6, 7, 8	otkelins2k 4256	. . . . . . 7 ⊢ (⟪{{{x}}}, ⟪A, B⟫⟫ ∈ Ins2k Sk ↔ ⟪{x}, B⟫ ∈ Sk )
14	10, 8	elssetk 4271	. . . . . . 7 ⊢ (⟪{x}, B⟫ ∈ Sk ↔ x ∈ B)
15	13, 14	bitri 240	. . . . . 6 ⊢ (⟪{{{x}}}, ⟪A, B⟫⟫ ∈ Ins2k Sk ↔ x ∈ B)
16	12, 15	anbi12i 678	. . . . 5 ⊢ ((⟪{{{x}}}, ⟪A, B⟫⟫ ∈ Ins3k Sk ∧ ⟪{{{x}}}, ⟪A, B⟫⟫ ∈ Ins2k Sk ) ↔ (x ∈ A ∧ x ∈ B))
17	5, 16	bitri 240	. . . 4 ⊢ (⟪{{{x}}}, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk ) ↔ (x ∈ A ∧ x ∈ B))
18	4, 17	bitri 240	. . 3 ⊢ (∃t(t = {{{x}}} ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )) ↔ (x ∈ A ∧ x ∈ B))
19	18	exbii 1582	. 2 ⊢ (∃x∃t(t = {{{x}}} ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )) ↔ ∃x(x ∈ A ∧ x ∈ B))
20		opkex 4114	. . . 4 ⊢ ⟪A, B⟫ ∈ V
21	20	elimak 4260	. . 3 ⊢ (⟪A, B⟫ ∈ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ↔ ∃t ∈ ℘1 ℘11c⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk ))
22		elpw121c 4149	. . . . . . 7 ⊢ (t ∈ ℘1℘11c ↔ ∃x t = {{{x}}})
23	22	anbi1i 676	. . . . . 6 ⊢ ((t ∈ ℘1℘11c ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )) ↔ (∃x t = {{{x}}} ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )))
24		19.41v 1901	. . . . . 6 ⊢ (∃x(t = {{{x}}} ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )) ↔ (∃x t = {{{x}}} ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )))
25	23, 24	bitr4i 243	. . . . 5 ⊢ ((t ∈ ℘1℘11c ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )) ↔ ∃x(t = {{{x}}} ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )))
26	25	exbii 1582	. . . 4 ⊢ (∃t(t ∈ ℘1℘11c ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )) ↔ ∃t∃x(t = {{{x}}} ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )))
27		df-rex 2621	. . . 4 ⊢ (∃t ∈ ℘1 ℘11c⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk ) ↔ ∃t(t ∈ ℘1℘11c ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )))
28		excom 1741	. . . 4 ⊢ (∃x∃t(t = {{{x}}} ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )) ↔ ∃t∃x(t = {{{x}}} ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )))
29	26, 27, 28	3bitr4i 268	. . 3 ⊢ (∃t ∈ ℘1 ℘11c⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk ) ↔ ∃x∃t(t = {{{x}}} ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )))
30	21, 29	bitri 240	. 2 ⊢ (⟪A, B⟫ ∈ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ↔ ∃x∃t(t = {{{x}}} ∧ ⟪t, ⟪A, B⟫⟫ ∈ ( Ins3k Sk ∩ Ins2k Sk )))
31		n0 3560	. . 3 ⊢ ((A ∩ B) ≠ ∅ ↔ ∃x x ∈ (A ∩ B))
32		elin 3220	. . . 4 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
33	32	exbii 1582	. . 3 ⊢ (∃x x ∈ (A ∩ B) ↔ ∃x(x ∈ A ∧ x ∈ B))
34	31, 33	bitri 240	. 2 ⊢ ((A ∩ B) ≠ ∅ ↔ ∃x(x ∈ A ∧ x ∈ B))
35	19, 30, 34	3bitr4i 268	1 ⊢ (⟪A, B⟫ ∈ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ↔ (A ∩ B) ≠ ∅)

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616  Vcvv 2860   ∩ cin 3209  ∅c0 3551  {csn 3738  ⟪copk 4058  1_cc1c 4135  ℘₁cpw1 4136   Ins2_k cins2k 4177   Ins3_k cins3k 4178   “_k cimak 4180   S_k cssetk 4184

This proof depends on 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 proof 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-rex 2621  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-pw 3725  df-sn 3742  df-pr 3743  df-opk 4059  df-1c 4137  df-pw1 4138  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-ssetk 4194

This theorem is used by:  dfaddc2  4382  nndisjeq  4430  ltfinex  4465  evenfinex  4504  oddfinex  4505  evenodddisjlem1  4516