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Theorem undefnel2 6547
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 6545 . 2  |-  -.  ~P U. S  e.  S
2 undefval 6546 . . 3  |-  ( S  e.  V  ->  ( Undef `  S )  =  ~P U. S )
32eleq1d 2502 . 2  |-  ( S  e.  V  ->  (
( Undef `  S )  e.  S  <->  ~P U. S  e.  S ) )
41, 3mtbiri 295 1  |-  ( S  e.  V  ->  -.  ( Undef `  S )  e.  S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1725   ~Pcpw 3799   U.cuni 4015   ` cfv 5454   Undefcund 6541
This theorem is referenced by:  undefnel  6548  riotaclbg  6589  riotaundb  6591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-undef 6543
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