Theorem iprc 4961

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 4187	. . 3 ⊢ ¬ V ∈ V
2		dmi 4907	. . . 4 ⊢ dom I = V
3	2	eleq1i 2272	. . 3 ⊢ (dom I ∈ V ↔ V ∈ V)
4	1, 3	mtbir 673	. 2 ⊢ ¬ dom I ∈ V
5		dmexg 4956	. 2 ⊢ ( I ∈ V → dom I ∈ V)
6	4, 5	mto 664	1 ⊢ ¬ I ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 2177  Vcvv 2773   I cid 4348  dom cdm 4688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-dm 4698  df-rn 4699

This theorem is referenced by: (None)