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Theorem undefnel 6540
Description: The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.)
Assertion
Ref Expression
undefnel  |-  ( S  e.  V  ->  ( Undef `  S )  e/  S )

Proof of Theorem undefnel
StepHypRef Expression
1 undefnel2 6539 . 2  |-  ( S  e.  V  ->  -.  ( Undef `  S )  e.  S )
2 df-nel 2601 . 2  |-  ( (
Undef `  S )  e/  S 
<->  -.  ( Undef `  S
)  e.  S )
31, 2sylibr 204 1  |-  ( S  e.  V  ->  ( Undef `  S )  e/  S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1725    e/ wnel 2599   ` cfv 5446   Undefcund 6533
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-undef 6535
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