ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iprc GIF version

Theorem iprc 4669
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
StepHypRef Expression
1 vprc 3946 . . 3 ¬ V ∈ V
2 dmi 4619 . . . 4 dom I = V
32eleq1i 2150 . . 3 (dom I ∈ V ↔ V ∈ V)
41, 3mtbir 629 . 2 ¬ dom I ∈ V
5 dmexg 4665 . 2 ( I ∈ V → dom I ∈ V)
64, 5mto 621 1 ¬ I ∈ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1436  Vcvv 2615   I cid 4089  dom cdm 4411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-dm 4421  df-rn 4422
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator