Theorem iprc 4765

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 4020	. . 3 ⊢ ¬ V ∈ V
2		dmi 4714	. . . 4 ⊢ dom I = V
3	2	eleq1i 2180	. . 3 ⊢ (dom I ∈ V ↔ V ∈ V)
4	1, 3	mtbir 643	. 2 ⊢ ¬ dom I ∈ V
5		dmexg 4761	. 2 ⊢ ( I ∈ V → dom I ∈ V)
6	4, 5	mto 634	1 ⊢ ¬ I ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 1463  Vcvv 2657   I cid 4170  dom cdm 4499

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-dm 4509  df-rn 4510

This theorem is referenced by: (None)