Theorem undefnel2 8295

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 8292	. 2 ⊢ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆
2		undefval 8294	. . 3 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆)
3	2	eleq1d 2845	. 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 2144  𝒫 cpw 4556  ∪ cuni 4866  ‘cfv 6538  Undefcund 8289

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-undef 8290

This theorem is used by:  undefnel  8296  riotaclbgBAD  40012