Theorem iprc 4930

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 4161	. . 3 ⊢ ¬ V ∈ V
2		dmi 4877	. . . 4 ⊢ dom I = V
3	2	eleq1i 2259	. . 3 ⊢ (dom I ∈ V ↔ V ∈ V)
4	1, 3	mtbir 672	. 2 ⊢ ¬ dom I ∈ V
5		dmexg 4926	. 2 ⊢ ( I ∈ V → dom I ∈ V)
6	4, 5	mto 663	1 ⊢ ¬ I ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 2164  Vcvv 2760   I cid 4319  dom cdm 4659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670

This theorem is referenced by: (None)