Theorem iprc 7899

Description: The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set, as in idcn 22743. (Contributed by NM, 1-Jan-2007.)

Assertion

Ref	Expression
iprc	⊢ ¬ I ∈ V

Proof of Theorem iprc

Step	Hyp	Ref	Expression
1		dmi 5919	. . 3 ⊢ dom I = V
2		vprc 5314	. . 3 ⊢ ¬ V ∈ V
3	1, 2	eqneltri 2853	. 2 ⊢ ¬ dom I ∈ V
4		dmexg 7889	. 2 ⊢ ( I ∈ V → dom I ∈ V)
5	3, 4	mto 196	1 ⊢ ¬ I ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 2107  Vcvv 3475   I cid 5572  dom cdm 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686

This theorem is referenced by: (None)