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 e. _V
ndisjrelk.2	|- B e. _V

Assertion

Ref	Expression
ndisjrelk	|- (<<A, B>> e. (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) <-> (A i^i 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}}} e. _V
2		opkeq1 4060	. . . . . 6 |- (t = {{{x}}} -> <<t, <<A, B>>>> = <<{{{x}}}, <<A, B>>>>)
3	2	eleq1d 2419	. . . . 5 |- (t = {{{x}}} -> (<<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk ) <-> <<{{{x}}}, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )))
4	1, 3	ceqsexv 2895	. . . 4 |- (E.t(t = {{{x}}} /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )) <-> <<{{{x}}}, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk ))
5		elin 3220	. . . . 5 |- (<<{{{x}}}, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk ) <-> (<<{{{x}}}, <<A, B>>>> e. Ins3k Sk /\ <<{{{x}}}, <<A, B>>>> e. Ins2k Sk ))
6		snex 4112	. . . . . . . 8 |- {x} e. _V
7		ndisjrelk.1	. . . . . . . 8 |- A e. _V
8		ndisjrelk.2	. . . . . . . 8 |- B e. _V
9	6, 7, 8	otkelins3k 4257	. . . . . . 7 |- (<<{{{x}}}, <<A, B>>>> e. Ins3k Sk <-> <<{x}, A>> e. Sk )
10		vex 2863	. . . . . . . 8 |- x e. _V
11	10, 7	elssetk 4271	. . . . . . 7 |- (<<{x}, A>> e. Sk <-> x e. A)
12	9, 11	bitri 240	. . . . . 6 |- (<<{{{x}}}, <<A, B>>>> e. Ins3k Sk <-> x e. A)
13	6, 7, 8	otkelins2k 4256	. . . . . . 7 |- (<<{{{x}}}, <<A, B>>>> e. Ins2k Sk <-> <<{x}, B>> e. Sk )
14	10, 8	elssetk 4271	. . . . . . 7 |- (<<{x}, B>> e. Sk <-> x e. B)
15	13, 14	bitri 240	. . . . . 6 |- (<<{{{x}}}, <<A, B>>>> e. Ins2k Sk <-> x e. B)
16	12, 15	anbi12i 678	. . . . 5 |- ((<<{{{x}}}, <<A, B>>>> e. Ins3k Sk /\ <<{{{x}}}, <<A, B>>>> e. Ins2k Sk ) <-> (x e. A /\ x e. B))
17	5, 16	bitri 240	. . . 4 |- (<<{{{x}}}, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk ) <-> (x e. A /\ x e. B))
18	4, 17	bitri 240	. . 3 |- (E.t(t = {{{x}}} /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )) <-> (x e. A /\ x e. B))
19	18	exbii 1582	. 2 |- (E.xE.t(t = {{{x}}} /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )) <-> E.x(x e. A /\ x e. B))
20		opkex 4114	. . . 4 |- <<A, B>> e. _V
21	20	elimak 4260	. . 3 |- (<<A, B>> e. (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) <-> E.t e. ~P1 ~P1 1c<<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk ))
22		elpw121c 4149	. . . . . . 7 |- (t e. ~P1 ~P1 1c <-> E.x t = {{{x}}})
23	22	anbi1i 676	. . . . . 6 |- ((t e. ~P1 ~P1 1c /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )) <-> (E.x t = {{{x}}} /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )))
24		19.41v 1901	. . . . . 6 |- (E.x(t = {{{x}}} /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )) <-> (E.x t = {{{x}}} /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )))
25	23, 24	bitr4i 243	. . . . 5 |- ((t e. ~P1 ~P1 1c /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )) <-> E.x(t = {{{x}}} /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )))
26	25	exbii 1582	. . . 4 |- (E.t(t e. ~P1 ~P1 1c /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )) <-> E.tE.x(t = {{{x}}} /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )))
27		df-rex 2621	. . . 4 |- (E.t e. ~P1 ~P1 1c<<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk ) <-> E.t(t e. ~P1 ~P1 1c /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )))
28		excom 1741	. . . 4 |- (E.xE.t(t = {{{x}}} /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )) <-> E.tE.x(t = {{{x}}} /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )))
29	26, 27, 28	3bitr4i 268	. . 3 |- (E.t e. ~P1 ~P1 1c<<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk ) <-> E.xE.t(t = {{{x}}} /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )))
30	21, 29	bitri 240	. 2 |- (<<A, B>> e. (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) <-> E.xE.t(t = {{{x}}} /\ <<t, <<A, B>>>> e. ( Ins3k Sk i^i Ins2k Sk )))
31		n0 3560	. . 3 |- ((A i^i B) =/= (/) <-> E.x x e. (A i^i B))
32		elin 3220	. . . 4 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
33	32	exbii 1582	. . 3 |- (E.x x e. (A i^i B) <-> E.x(x e. A /\ x e. B))
34	31, 33	bitri 240	. 2 |- ((A i^i B) =/= (/) <-> E.x(x e. A /\ x e. B))
35	19, 30, 34	3bitr4i 268	1 |- (<<A, B>> e. (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) <-> (A i^i B) =/= (/))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710   =/= wne 2517  E.wrex 2616  _Vcvv 2860   i^i cin 3209  (/)c0 3551  {csn 3738  <<copk 4058  1_cc1c 4135  ~P₁ cpw1 4136   Ins2_k cins2k 4177   Ins3_k cins3k 4178  "_kcimak 4180   S_k cssetk 4184

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-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 referenced by:  dfaddc2  4382  nndisjeq  4430  ltfinex  4465  evenfinex  4504  oddfinex  4505  evenodddisjlem1  4516