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

Step	Hyp	Ref	Expression
1		vprc 3946	. . 3 ⊢ ¬ V ∈ V
2		dmi 4619	. . . 4 ⊢ dom I = V
3	2	eleq1i 2150	. . 3 ⊢ (dom I ∈ V ↔ V ∈ V)
4	1, 3	mtbir 629	. 2 ⊢ ¬ dom I ∈ V
5		dmexg 4665	. 2 ⊢ ( I ∈ V → dom I ∈ V)
6	4, 5	mto 621	1 ⊢ ¬ I ∈ V

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)