Theorem undefnel2 7942

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 7940	. 2 ⊢ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆
2		undefval 7941	. . 3 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆)
3	2	eleq1d 2897	. 2 ⊢ (𝑆 ∈ 𝑉 → ((Undef‘𝑆) ∈ 𝑆 ↔ 𝒫 ∪ 𝑆 ∈ 𝑆))
4	1, 3	mtbiri 329	1 ⊢ (𝑆 ∈ 𝑉 → ¬ (Undef‘𝑆) ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2110  𝒫 cpw 4538  ∪ cuni 4837  ‘cfv 6354  Undefcund 7937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-undef 7938

This theorem is referenced by:  undefnel  7943  riotaclbgBAD  36089