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Theorem nfunv 5879
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 5103 . . 3 ¬ ∅ ∈ (V × V)
2 0ex 4750 . . . 4 ∅ ∈ V
3 df-rel 5081 . . . . . 6 (Rel V ↔ V ⊆ (V × V))
43biimpi 206 . . . . 5 (Rel V → V ⊆ (V × V))
54sseld 3582 . . . 4 (Rel V → (∅ ∈ V → ∅ ∈ (V × V)))
62, 5mpi 20 . . 3 (Rel V → ∅ ∈ (V × V))
71, 6mto 188 . 2 ¬ Rel V
8 funrel 5864 . 2 (Fun V → Rel V)
97, 8mto 188 1 ¬ Fun V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 1987  Vcvv 3186  wss 3555  c0 3891   × cxp 5072  Rel wrel 5079  Fun wfun 5841
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
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-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-opab 4674  df-xp 5080  df-rel 5081  df-fun 5849
This theorem is referenced by: (None)
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