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Theorem iswatN 30805
Description: The predicate "is a W atom" (corresponding to fiducial atom  D). (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
watomfval.a  |-  A  =  ( Atoms `  K )
watomfval.p  |-  P  =  ( _|_ P `  K )
watomfval.w  |-  W  =  ( WAtoms `  K )
Assertion
Ref Expression
iswatN  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( P  e.  ( W `  D )  <-> 
( P  e.  A  /\  -.  P  e.  ( ( _|_ P `  K ) `  { D } ) ) ) )

Proof of Theorem iswatN
StepHypRef Expression
1 watomfval.a . . . 4  |-  A  =  ( Atoms `  K )
2 watomfval.p . . . 4  |-  P  =  ( _|_ P `  K )
3 watomfval.w . . . 4  |-  W  =  ( WAtoms `  K )
41, 2, 3watvalN 30804 . . 3  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( W `  D
)  =  ( A 
\  ( ( _|_
P `  K ) `  { D } ) ) )
54eleq2d 2363 . 2  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( P  e.  ( W `  D )  <-> 
P  e.  ( A 
\  ( ( _|_
P `  K ) `  { D } ) ) ) )
6 eldif 3175 . 2  |-  ( P  e.  ( A  \ 
( ( _|_ P `  K ) `  { D } ) )  <->  ( P  e.  A  /\  -.  P  e.  ( ( _|_ P `  K ) `  { D } ) ) )
75, 6syl6bb 252 1  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( P  e.  ( W `  D )  <-> 
( P  e.  A  /\  -.  P  e.  ( ( _|_ P `  K ) `  { D } ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    \ cdif 3162   {csn 3653   ` cfv 5271   Atomscatm 30075   _|_
PcpolN 30713   WAtomscwpointsN 30797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-watsN 30801
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