Theorem nfunv 5921

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 5143	. . 3 ⊢ ¬ ∅ ∈ (V × V)
2		0ex 4790	. . . 4 ⊢ ∅ ∈ V
3		df-rel 5121	. . . . . 6 ⊢ (Rel V ↔ V ⊆ (V × V))
4	3	biimpi 206	. . . . 5 ⊢ (Rel V → V ⊆ (V × V))
5	4	sseld 3602	. . . 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 5905	. 2 ⊢ (Fun V → Rel V)
9	7, 8	mto 188	1 ⊢ ¬ Fun V

Syntax hints:  ¬ wn 3   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  ∅c0 3915   × cxp 5112  Rel wrel 5119  Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121  df-fun 5890

This theorem is referenced by: (None)