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Theorem supp0prc 7466
 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 7464 . 2 supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
21mpt2ndm0 7040 1 (¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  {crab 3054  Vcvv 3340  ∅c0 4058  {csn 4321  dom cdm 5266   “ cima 5269  (class class class)co 6813   supp csupp 7463 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-dm 5276  df-iota 6012  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-supp 7464 This theorem is referenced by:  suppssdm  7476  suppun  7483  extmptsuppeq  7487  funsssuppss  7490  fczsupp0  7493  suppss  7494  suppssov1  7496  suppss2  7498  suppssfv  7500  supp0cosupp0  7503  imacosupp  7504  fsuppun  8459
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