Theorem iprc 7930

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 23587. (Contributed by NM, 1-Jan-2007.)

Assertion

Ref	Expression
iprc	⊢ ¬ I ∈ V

Proof of Theorem iprc

Step	Hyp	Ref	Expression
1		dmi 5907	. . 3 ⊢ dom I = V
2		vprc 5277	. . 3 ⊢ ¬ V ∈ V
3	1, 2	eqneltri 2882	. 2 ⊢ ¬ dom I ∈ V
4		dmexg 7920	. 2 ⊢ ( I ∈ V → dom I ∈ V)
5	3, 4	mto 200	1 ⊢ ¬ I ∈ V

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2146  Vcvv 3453   I cid 5548  dom cdm 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665

This theorem is used by: (None)