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Theorem nfunv 5835
 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 5067 . . 3 ¬ ∅ ∈ (V × V)
2 0ex 4718 . . . 4 ∅ ∈ V
3 df-rel 5045 . . . . . 6 (Rel V ↔ V ⊆ (V × V))
43biimpi 205 . . . . 5 (Rel V → V ⊆ (V × V))
54sseld 3567 . . . 4 (Rel V → (∅ ∈ V → ∅ ∈ (V × V)))
62, 5mpi 20 . . 3 (Rel V → ∅ ∈ (V × V))
71, 6mto 187 . 2 ¬ Rel V
8 funrel 5821 . 2 (Fun V → Rel V)
97, 8mto 187 1 ¬ Fun V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874   × cxp 5036  Rel wrel 5043  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045  df-fun 5806 This theorem is referenced by: (None)
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