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Theorem undefnel 6908
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 6907 . 2  |-  ( S  e.  V  ->  -.  ( Undef `  S )  e.  S )
2 df-nel 2651 . 2  |-  ( (
Undef `  S )  e/  S 
<->  -.  ( Undef `  S
)  e.  S )
31, 2sylibr 212 1  |-  ( S  e.  V  ->  ( Undef `  S )  e/  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1758    e/ wnel 2649   ` cfv 5527   Undefcund 6902
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 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-undef 6903
This theorem is referenced by: (None)
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