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Theorem undefne0 7020
Description: The undefined value generated from a set is not empty. (Contributed by NM, 3-Sep-2018.)
Assertion
Ref Expression
undefne0  |-  ( S  e.  V  ->  ( Undef `  S )  =/=  (/) )

Proof of Theorem undefne0
StepHypRef Expression
1 undefval 7017 . 2  |-  ( S  e.  V  ->  ( Undef `  S )  =  ~P U. S )
2 pwne0 4623 . . 3  |-  ~P U. S  =/=  (/)
32a1i 11 . 2  |-  ( S  e.  V  ->  ~P U. S  =/=  (/) )
41, 3eqnetrd 2760 1  |-  ( S  e.  V  ->  ( Undef `  S )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767    =/= wne 2662   (/)c0 3790   ~Pcpw 4016   U.cuni 4251   ` cfv 5594   Undefcund 7013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-undef 7014
This theorem is referenced by: (None)
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