Theorem iprc 5051

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 4265	. . 3 ⊢ ¬ V ∈ V
2		dmi 4996	. . . 4 ⊢ dom I = V
3	2	eleq1i 2304	. . 3 ⊢ (dom I ∈ V ↔ V ∈ V)
4	1, 3	mtbir 682	. 2 ⊢ ¬ dom I ∈ V
5		dmexg 5046	. 2 ⊢ ( I ∈ V → dom I ∈ V)
6	4, 5	mto 672	1 ⊢ ¬ I ∈ V

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2209  Vcvv 2821   I cid 4433  dom cdm 4774

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-dm 4784  df-rn 4785

This theorem is used by: (None)