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Theorem undefnel2 7448
 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 (𝑆𝑉 → ¬ (Undef‘𝑆) ∈ 𝑆)

Proof of Theorem undefnel2
StepHypRef Expression
1 pwuninel 7446 . 2 ¬ 𝒫 𝑆𝑆
2 undefval 7447 . . 3 (𝑆𝑉 → (Undef‘𝑆) = 𝒫 𝑆)
32eleq1d 2715 . 2 (𝑆𝑉 → ((Undef‘𝑆) ∈ 𝑆 ↔ 𝒫 𝑆𝑆))
41, 3mtbiri 316 1 (𝑆𝑉 → ¬ (Undef‘𝑆) ∈ 𝑆)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2030  𝒫 cpw 4191  ∪ cuni 4468  ‘cfv 5926  Undefcund 7443 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-undef 7444 This theorem is referenced by:  undefnel  7449  riotaclbgBAD  34558
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