Theorem undefnel 8164

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 8163	. 2 ⊢ (𝑆 ∈ 𝑉 → ¬ (Undef‘𝑆) ∈ 𝑆)
2		df-nel 3047	. 2 ⊢ ((Undef‘𝑆) ∉ 𝑆 ↔ ¬ (Undef‘𝑆) ∈ 𝑆)
3	1, 2	sylibr 233	1 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) ∉ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2105   ∉ wnel 3046  ‘cfv 6479  Undefcund 8158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-undef 8159

This theorem is referenced by: (None)