Theorem undefnel 8230

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 8229	. 2 ⊢ (𝑆 ∈ 𝑉 → ¬ (Undef‘𝑆) ∈ 𝑆)
2		df-nel 3038	. 2 ⊢ ((Undef‘𝑆) ∉ 𝑆 ↔ ¬ (Undef‘𝑆) ∈ 𝑆)
3	1, 2	sylibr 234	1 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) ∉ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114   ∉ wnel 3037  ‘cfv 6500  Undefcund 8224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-undef 8225

This theorem is referenced by: (None)