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Theorem supp0prc 7820
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 ∈ V ∧ 𝑍 ∈ 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 7818 . 2 supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
21mpondm0 7370 1 (¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2112  wne 2990  {crab 3113  Vcvv 3444  c0 4246  {csn 4528  dom cdm 5523  cima 5526  (class class class)co 7139   supp csupp 7817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-dm 5533  df-iota 6287  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-supp 7818
This theorem is referenced by:  suppssdm  7830  suppun  7837  extmptsuppeq  7841  funsssuppss  7843  fczsupp0  7846  suppss  7847  suppssov1  7849  suppss2  7851  suppssfv  7853  suppco  7857  supp0cosupp0OLD  7860  imacosuppOLD  7862  fsuppun  8840
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