Theorem supp0prc 8103

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 8101	. 2 ⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
2	1	mpondm0 7596	1 ⊢ (¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  {crab 3391  Vcvv 3431  ∅c0 4261  {csn 4555  dom cdm 5618   “ cima 5621  (class class class)co 7356   supp csupp 8100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-dm 5628  df-iota 6441  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-supp 8101

This theorem is referenced by:  suppssdm  8117  suppun  8124  extmptsuppeq  8128  funsssuppss  8130  fczsupp0  8133  suppss  8134  suppssov1  8137  suppssov2  8138  suppss2  8140  suppssfv  8142  suppco  8146  fsuppun  9290