Theorem iprc 6966

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 20809. (Contributed by NM, 1-Jan-2007.)

Assertion

Ref	Expression
iprc	⊢ ¬ I ∈ V

Proof of Theorem iprc

Step	Hyp	Ref	Expression
1		vprc 4715	. . 3 ⊢ ¬ V ∈ V
2		dmi 5244	. . . 4 ⊢ dom I = V
3	2	eleq1i 2674	. . 3 ⊢ (dom I ∈ V ↔ V ∈ V)
4	1, 3	mtbir 311	. 2 ⊢ ¬ dom I ∈ V
5		dmexg 6962	. 2 ⊢ ( I ∈ V → dom I ∈ V)
6	4, 5	mto 186	1 ⊢ ¬ I ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 1975  Vcvv 3168   I cid 4934  dom cdm 5024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824  ax-un 6820

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-dm 5034  df-rn 5035

This theorem is referenced by: (None)