Theorem supp0prc 8142

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 8140	. 2 ⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
2	1	mpondm0 7629	1 ⊢ (¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {crab 3405  Vcvv 3447  ∅c0 4296  {csn 4589  dom cdm 5638   “ cima 5641  (class class class)co 7387   supp csupp 8139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-dm 5648  df-iota 6464  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-supp 8140

This theorem is referenced by:  suppssdm  8156  suppun  8163  extmptsuppeq  8167  funsssuppss  8169  fczsupp0  8172  suppss  8173  suppssov1  8176  suppssov2  8177  suppss2  8179  suppssfv  8181  suppco  8185  fsuppun  9338