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Theorem undefne0 7023
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 7020 . 2  |-  ( S  e.  V  ->  ( Undef `  S )  =  ~P U. S )
2 pwne0 4572 . . 3  |-  ~P U. S  =/=  (/)
32a1i 11 . 2  |-  ( S  e.  V  ->  ~P U. S  =/=  (/) )
41, 3eqnetrd 2690 1  |-  ( S  e.  V  ->  ( Undef `  S )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1886    =/= wne 2621   (/)c0 3730   ~Pcpw 3950   U.cuni 4197   ` cfv 5581   Undefcund 7016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589  df-undef 7017
This theorem is referenced by: (None)
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