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

Step	Hyp	Ref	Expression
1		0nelxp 5300	. . 3 ⊢ ¬ ∅ ∈ (V × V)
2		0ex 4942	. . . 4 ⊢ ∅ ∈ V
3		df-rel 5273	. . . . . 6 ⊢ (Rel V ↔ V ⊆ (V × V))
4	3	biimpi 206	. . . . 5 ⊢ (Rel V → V ⊆ (V × V))
5	4	sseld 3743	. . . 4 ⊢ (Rel V → (∅ ∈ V → ∅ ∈ (V × V)))
6	2, 5	mpi 20	. . 3 ⊢ (Rel V → ∅ ∈ (V × V))
7	1, 6	mto 188	. 2 ⊢ ¬ Rel V
8		funrel 6066	. 2 ⊢ (Fun V → Rel V)
9	7, 8	mto 188	1 ⊢ ¬ Fun V

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)