Theorem iprc 7909

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 23483. (Contributed by NM, 1-Jan-2007.)

Assertion

Ref	Expression
iprc	⊢ ¬ I ∈ V

Proof of Theorem iprc

Step	Hyp	Ref	Expression
1		dmi 5905	. . 3 ⊢ dom I = V
2		vprc 5277	. . 3 ⊢ ¬ V ∈ V
3	1, 2	eqneltri 2879	. 2 ⊢ ¬ dom I ∈ V
4		dmexg 7899	. 2 ⊢ ( I ∈ V → dom I ∈ V)
5	3, 4	mto 200	1 ⊢ ¬ I ∈ V

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2145  Vcvv 3450   I cid 5549  dom cdm 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by: (None)