Theorem iprc 7908

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 23383. (Contributed by NM, 1-Jan-2007.)

Assertion

Ref	Expression
iprc	⊢ ¬ I ∈ V

Proof of Theorem iprc

Step	Hyp	Ref	Expression
1		dmi 5912	. . 3 ⊢ dom I = V
2		vprc 5285	. . 3 ⊢ ¬ V ∈ V
3	1, 2	eqneltri 2888	. 2 ⊢ ¬ dom I ∈ V
4		dmexg 7898	. 2 ⊢ ( I ∈ V → dom I ∈ V)
5	3, 4	mto 200	1 ⊢ ¬ I ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 2149  Vcvv 3463   I cid 5556  dom cdm 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-rn 5673

This theorem is referenced by: (None)