Theorem iprc 7915

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 23211. (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 5295	. . 3 ⊢ ¬ V ∈ V
3	1, 2	eqneltri 2852	. 2 ⊢ ¬ dom I ∈ V
4		dmexg 7905	. 2 ⊢ ( I ∈ V → dom I ∈ V)
5	3, 4	mto 197	1 ⊢ ¬ I ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 2107  Vcvv 3463   I cid 5557  dom cdm 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-dm 5675  df-rn 5676

This theorem is referenced by: (None)