Theorem supp0prc 7951

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.

Step	Hyp	Ref	Expression
1		df-supp 7949	. 2 ⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
2	1	mpondm0 7488	1 ⊢ (¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  {crab 3067  Vcvv 3422  ∅c0 4253  {csn 4558  dom cdm 5580   “ cima 5583  (class class class)co 7255   supp csupp 7948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-dm 5590  df-iota 6376  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-supp 7949

This theorem is referenced by:  suppssdm  7964  suppun  7971  extmptsuppeq  7975  funsssuppss  7977  fczsupp0  7980  suppss  7981  suppssOLD  7982  suppssov1  7985  suppss2  7987  suppssfv  7989  suppco  7993  fsuppun  9077