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

Theorem nfunv 6082
Description: The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
Assertion
Ref Expression
nfunv ¬ Fun V

Proof of Theorem nfunv
StepHypRef Expression
1 0nelxp 5300 . . 3 ¬ ∅ ∈ (V × V)
2 0ex 4942 . . . 4 ∅ ∈ V
3 df-rel 5273 . . . . . 6 (Rel V ↔ V ⊆ (V × V))
43biimpi 206 . . . . 5 (Rel V → V ⊆ (V × V))
54sseld 3743 . . . 4 (Rel V → (∅ ∈ V → ∅ ∈ (V × V)))
62, 5mpi 20 . . 3 (Rel V → ∅ ∈ (V × V))
71, 6mto 188 . 2 ¬ Rel V
8 funrel 6066 . 2 (Fun V → Rel V)
97, 8mto 188 1 ¬ Fun V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2139  Vcvv 3340  wss 3715  c0 4058   × cxp 5264  Rel wrel 5271  Fun wfun 6043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272  df-rel 5273  df-fun 6051
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator