Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjnf Structured version   Unicode version

Theorem disjnf 27256
 Description: In case is not free in , disjointness is not so interesting since it reduces to cases where is a singleton. (Google Groups discussion with Peter Masza) (Contributed by Thierry Arnoux, 26-Jul-2018.)
Assertion
Ref Expression
disjnf Disj
Distinct variable groups:   ,   ,

Proof of Theorem disjnf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqidd 2468 . . . . 5
21disjor 4437 . . . 4 Disj
3 orcom 387 . . . . . . 7
43ralbii 2898 . . . . . 6
5 r19.32v 3012 . . . . . 6
64, 5bitri 249 . . . . 5
76ralbii 2898 . . . 4
8 r19.32v 3012 . . . 4
92, 7, 83bitri 271 . . 3 Disj
10 inidm 3712 . . . . 5
1110eqeq1i 2474 . . . 4
1211orbi1i 520 . . 3
139, 12bitri 249 . 2 Disj
14 moel 27205 . . 3
1514orbi2i 519 . 2
1613, 15bitr4i 252 1 Disj
 Colors of variables: wff setvar class Syntax hints:   wb 184   wo 368   wceq 1379   wcel 1767  wmo 2276  wral 2817   cin 3480  c0 3790  Disj wdisj 4423 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rmo 2825  df-v 3120  df-dif 3484  df-in 3488  df-nul 3791  df-disj 4424 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator