Theorem iprc 4802

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

Assertion

Ref	Expression
iprc	⊢ ¬ I ∈ V

Proof of Theorem iprc

Step	Hyp	Ref	Expression
1		vprc 4055	. . 3 ⊢ ¬ V ∈ V
2		dmi 4749	. . . 4 ⊢ dom I = V
3	2	eleq1i 2203	. . 3 ⊢ (dom I ∈ V ↔ V ∈ V)
4	1, 3	mtbir 660	. 2 ⊢ ¬ dom I ∈ V
5		dmexg 4798	. 2 ⊢ ( I ∈ V → dom I ∈ V)
6	4, 5	mto 651	1 ⊢ ¬ I ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 1480  Vcvv 2681   I cid 4205  dom cdm 4534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545

This theorem is referenced by: (None)