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Theorem disjnf 28017
 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 inidm 3668 . . . 4
21eqeq1i 2427 . . 3
32orbi1i 522 . 2
4 eqidd 2421 . . . 4
54disjor 4402 . . 3 Disj
6 orcom 388 . . . . . 6
76ralbii 2854 . . . . 5
8 r19.32v 2972 . . . . 5
97, 8bitri 252 . . . 4
109ralbii 2854 . . 3
11 r19.32v 2972 . . 3
125, 10, 113bitri 274 . 2 Disj
13 moel 27951 . . 3
1413orbi2i 521 . 2
153, 12, 143bitr4i 280 1 Disj
 Colors of variables: wff setvar class Syntax hints:   wb 187   wo 369   wceq 1437   wcel 1867  wmo 2264  wral 2773   cin 3432  c0 3758  Disj wdisj 4388 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rmo 2781  df-v 3080  df-dif 3436  df-in 3440  df-nul 3759  df-disj 4389 This theorem is referenced by: (None)
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