MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supp0prc Structured version   Unicode version

Theorem supp0prc 6924
Description: The support of a class is empty if either the class or the "zero" is a proper class. . (Contributed by AV, 28-May-2019.)
Assertion
Ref Expression
supp0prc  |-  ( -.  ( X  e.  _V  /\  Z  e.  _V )  ->  ( X supp  Z )  =  (/) )

Proof of Theorem supp0prc
Dummy variables  x  z  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-supp 6922 . 2  |- supp  =  ( x  e.  _V , 
z  e.  _V  |->  { i  e.  dom  x  |  ( x " { i } )  =/=  { z } } )
21mpt2ndm0 6520 1  |-  ( -.  ( X  e.  _V  /\  Z  e.  _V )  ->  ( X supp  Z )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   {crab 2779   _Vcvv 3081   (/)c0 3761   {csn 3996   dom cdm 4849   "cima 4852  (class class class)co 6301   supp csupp 6921
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 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-xp 4855  df-dm 4859  df-iota 5561  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-supp 6922
This theorem is referenced by:  suppssdm  6934  suppun  6942  extmptsuppeq  6946  funsssuppss  6948  fczsupp0  6951  suppss  6952  suppssov1  6954  suppss2  6956  suppssfv  6958  supp0cosupp0  6961  imacosupp  6962  fsuppun  7904
  Copyright terms: Public domain W3C validator