Theorem supp0prc 8180

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 8178	. 2 ⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
2	1	mpondm0 7661	1 ⊢ (¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  {crab 3413  Vcvv 3451  ∅c0 4279  {csn 4584  dom cdm 5651   “ cima 5654  (class class class)co 7420   supp csupp 8177

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-dm 5661  df-iota 6494  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-supp 8178

This theorem is used by:  suppssdm  8194  suppun  8201  extmptsuppeq  8205  funsssuppss  8207  fczsupp0  8210  suppss  8211  suppssov1  8214  suppssov2  8215  suppss2  8217  suppssfv  8219  suppco  8223  fsuppun  9379