MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supp0prc Structured version   Visualization version   GIF version

Theorem supp0prc 7258
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 7256 . 2 supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
21mpt2ndm0 6840 1 (¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  {crab 2912  Vcvv 3190  c0 3897  {csn 4155  dom cdm 5084  cima 5087  (class class class)co 6615   supp csupp 7255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-dm 5094  df-iota 5820  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-supp 7256
This theorem is referenced by:  suppssdm  7268  suppun  7275  extmptsuppeq  7279  funsssuppss  7281  fczsupp0  7284  suppss  7285  suppssov1  7287  suppss2  7289  suppssfv  7291  supp0cosupp0  7294  imacosupp  7295  fsuppun  8254
  Copyright terms: Public domain W3C validator