Theorem undefnel2 6998

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 6996	. 2 |- -. ~P U. S e. S
2		undefval 6997	. . 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 301	1 |- ( S e. V -> -. ( Undef ` S ) e. S )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1823   ~Pcpw 3999   U.cuni 4235   ` cfv 5570   Undefcund 6993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-undef 6994

This theorem is referenced by:  undefnel  6999  riotaclbgBAD  35082