Theorem undefnel2 8318

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 8316	. 2 ⊢ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆
2		undefval 8317	. . 3 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆)
3	2	eleq1d 2829	. 2 ⊢ (𝑆 ∈ 𝑉 → ((Undef‘𝑆) ∈ 𝑆 ↔ 𝒫 ∪ 𝑆 ∈ 𝑆))
4	1, 3	mtbiri 327	1 ⊢ (𝑆 ∈ 𝑉 → ¬ (Undef‘𝑆) ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2108  𝒫 cpw 4622  ∪ cuni 4931  ‘cfv 6573  Undefcund 8313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-undef 8314

This theorem is referenced by:  undefnel  8319  riotaclbgBAD  38910