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Theorem undefnel 7933
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 (𝑆𝑉 → (Undef‘𝑆) ∉ 𝑆)

Proof of Theorem undefnel
StepHypRef Expression
1 undefnel2 7932 . 2 (𝑆𝑉 → ¬ (Undef‘𝑆) ∈ 𝑆)
2 df-nel 3121 . 2 ((Undef‘𝑆) ∉ 𝑆 ↔ ¬ (Undef‘𝑆) ∈ 𝑆)
31, 2sylibr 235 1 (𝑆𝑉 → (Undef‘𝑆) ∉ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2105  wnel 3120  cfv 6348  Undefcund 7927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-undef 7928
This theorem is referenced by: (None)
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