Theorem undefnel2 6909

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 e. V -> -. ( Undef ` S ) e. S )

Proof of Theorem undefnel2

Step	Hyp	Ref	Expression
1		pwuninel 6907	. 2 |- -. ~P U. S e. S
2		undefval 6908	. . 3 |- ( S e. V -> ( Undef ` S ) = ~P U. S )
3	2	eleq1d 2523	. 2 |- ( S e. V -> ( ( Undef ` S ) e. S <-> ~P U. S e. S ) )
4	1, 3	mtbiri 303	1 |- ( S e. V -> -. ( Undef ` S ) e. S )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1758   ~Pcpw 3971   U.cuni 4202   ` cfv 5529   Undefcund 6904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-undef 6905

This theorem is referenced by:  undefnel  6910  riotaclbgBAD  32963  riotaclbBAD  32964