MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0nelfun Structured version   Visualization version   GIF version

Theorem 0nelfun 6574
Description: A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)
Assertion
Ref Expression
0nelfun (Fun 𝑅 → ∅ ∉ 𝑅)

Proof of Theorem 0nelfun
StepHypRef Expression
1 funrel 6573 . 2 (Fun 𝑅 → Rel 𝑅)
2 0nelrel 5741 . 2 (Rel 𝑅 → ∅ ∉ 𝑅)
31, 2syl 17 1 (Fun 𝑅 → ∅ ∉ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnel 3042  c0 4324  Rel wrel 5685  Fun wfun 6545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-nel 3043  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-opab 5213  df-xp 5686  df-rel 5687  df-fun 6553
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator