Theorem undefnel 8274

Description: The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.)

Assertion

Ref	Expression
undefnel	⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) ∉ 𝑆)

Proof of Theorem undefnel

Step	Hyp	Ref	Expression
1		undefnel2 8273	. 2 ⊢ (𝑆 ∈ 𝑉 → ¬ (Undef‘𝑆) ∈ 𝑆)
2		df-nel 3063	. 2 ⊢ ((Undef‘𝑆) ∉ 𝑆 ↔ ¬ (Undef‘𝑆) ∈ 𝑆)
3	1, 2	sylibr 237	1 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) ∉ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2141   ∉ wnel 3062  ‘cfv 6536  Undefcund 8267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-undef 8268

This theorem is referenced by: (None)