Theorem undefnel2 8280

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 ∈ 𝑉 → ¬ (Undef‘𝑆) ∈ 𝑆)

Proof of Theorem undefnel2

Step	Hyp	Ref	Expression
1		pwuninel 8277	. 2 ⊢ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆
2		undefval 8279	. . 3 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆)
3	2	eleq1d 2850	. 2 ⊢ (𝑆 ∈ 𝑉 → ((Undef‘𝑆) ∈ 𝑆 ↔ 𝒫 ∪ 𝑆 ∈ 𝑆))
4	1, 3	mtbiri 330	1 ⊢ (𝑆 ∈ 𝑉 → ¬ (Undef‘𝑆) ∈ 𝑆)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2146  𝒫 cpw 4564  ∪ cuni 4874  ‘cfv 6540  Undefcund 8274

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-undef 8275

This theorem is used by:  undefnel  8281  riotaclbgBAD  39790