MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  undefnel2 Structured version   Unicode version

Theorem undefnel2 6909
Description: The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.)
Assertion
Ref Expression
undefnel2  |-  ( S  e.  V  ->  -.  ( Undef `  S )  e.  S )

Proof of Theorem undefnel2
StepHypRef Expression
1 pwuninel 6907 . 2  |-  -.  ~P U. S  e.  S
2 undefval 6908 . . 3  |-  ( S  e.  V  ->  ( Undef `  S )  =  ~P U. S )
32eleq1d 2523 . 2  |-  ( S  e.  V  ->  (
( Undef `  S )  e.  S  <->  ~P U. S  e.  S ) )
41, 3mtbiri 303 1  |-  ( S  e.  V  ->  -.  ( Undef `  S )  e.  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1758   ~Pcpw 3971   U.cuni 4202   ` cfv 5529   Undefcund 6904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-undef 6905
This theorem is referenced by:  undefnel  6910  riotaclbgBAD  32963  riotaclbBAD  32964
  Copyright terms: Public domain W3C validator