New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  ndisjrelk GIF version

Theorem ndisjrelk 4323
 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 SkIns2k Sk ) “k 111c) ↔ (AB) ≠ )

Proof of Theorem ndisjrelk
Dummy variables x t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4111 . . . . 5 {{{x}}} V
2 opkeq1 4059 . . . . . 6 (t = {{{x}}} → ⟪t, ⟪A, B⟫⟫ = ⟪{{{x}}}, ⟪A, B⟫⟫)
32eleq1d 2419 . . . . 5 (t = {{{x}}} → (⟪t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk ) ↔ ⟪{{{x}}}, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )))
41, 3ceqsexv 2894 . . . 4 (t(t = {{{x}}} t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )) ↔ ⟪{{{x}}}, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk ))
5 elin 3219 . . . . 5 (⟪{{{x}}}, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk ) ↔ (⟪{{{x}}}, ⟪A, B⟫⟫ Ins3k Sk ⟪{{{x}}}, ⟪A, B⟫⟫ Ins2k Sk ))
6 snex 4111 . . . . . . . 8 {x} V
7 ndisjrelk.1 . . . . . . . 8 A V
8 ndisjrelk.2 . . . . . . . 8 B V
96, 7, 8otkelins3k 4256 . . . . . . 7 (⟪{{{x}}}, ⟪A, B⟫⟫ Ins3k Sk ↔ ⟪{x}, A Sk )
10 vex 2862 . . . . . . . 8 x V
1110, 7elssetk 4270 . . . . . . 7 (⟪{x}, A Skx A)
129, 11bitri 240 . . . . . 6 (⟪{{{x}}}, ⟪A, B⟫⟫ Ins3k Skx A)
136, 7, 8otkelins2k 4255 . . . . . . 7 (⟪{{{x}}}, ⟪A, B⟫⟫ Ins2k Sk ↔ ⟪{x}, B Sk )
1410, 8elssetk 4270 . . . . . . 7 (⟪{x}, B Skx B)
1513, 14bitri 240 . . . . . 6 (⟪{{{x}}}, ⟪A, B⟫⟫ Ins2k Skx B)
1612, 15anbi12i 678 . . . . 5 ((⟪{{{x}}}, ⟪A, B⟫⟫ Ins3k Sk ⟪{{{x}}}, ⟪A, B⟫⟫ Ins2k Sk ) ↔ (x A x B))
175, 16bitri 240 . . . 4 (⟪{{{x}}}, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk ) ↔ (x A x B))
184, 17bitri 240 . . 3 (t(t = {{{x}}} t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )) ↔ (x A x B))
1918exbii 1582 . 2 (xt(t = {{{x}}} t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )) ↔ x(x A x B))
20 opkex 4113 . . . 4 A, B V
2120elimak 4259 . . 3 (⟪A, B (( Ins3k SkIns2k Sk ) “k 111c) ↔ t 1 11ct, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk ))
22 elpw121c 4148 . . . . . . 7 (t 111cx t = {{{x}}})
2322anbi1i 676 . . . . . 6 ((t 111c t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )) ↔ (x t = {{{x}}} t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )))
24 19.41v 1901 . . . . . 6 (x(t = {{{x}}} t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )) ↔ (x t = {{{x}}} t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )))
2523, 24bitr4i 243 . . . . 5 ((t 111c t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )) ↔ x(t = {{{x}}} t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )))
2625exbii 1582 . . . 4 (t(t 111c t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )) ↔ tx(t = {{{x}}} t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )))
27 df-rex 2620 . . . 4 (t 1 11ct, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk ) ↔ t(t 111c t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )))
28 excom 1741 . . . 4 (xt(t = {{{x}}} t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )) ↔ tx(t = {{{x}}} t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )))
2926, 27, 283bitr4i 268 . . 3 (t 1 11ct, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk ) ↔ xt(t = {{{x}}} t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )))
3021, 29bitri 240 . 2 (⟪A, B (( Ins3k SkIns2k Sk ) “k 111c) ↔ xt(t = {{{x}}} t, ⟪A, B⟫⟫ ( Ins3k SkIns2k Sk )))
31 n0 3559 . . 3 ((AB) ≠ x x (AB))
32 elin 3219 . . . 4 (x (AB) ↔ (x A x B))
3332exbii 1582 . . 3 (x x (AB) ↔ x(x A x B))
3431, 33bitri 240 . 2 ((AB) ≠ x(x A x B))
3519, 30, 343bitr4i 268 1 (⟪A, B (( Ins3k SkIns2k Sk ) “k 111c) ↔ (AB) ≠ )
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  Vcvv 2859   ∩ cin 3208  ∅c0 3550  {csn 3737  ⟪copk 4057  1cc1c 4134  ℘1cpw1 4135   Ins2k cins2k 4176   Ins3k cins3k 4177   “k cimak 4179   Sk cssetk 4183 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-rex 2620  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-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-ssetk 4193 This theorem is referenced by:  dfaddc2  4381  nndisjeq  4429  ltfinex  4464  evenfinex  4503  oddfinex  4504  evenodddisjlem1  4515
 Copyright terms: Public domain W3C validator