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Theorem iswatN 30183
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 30182 . . 3  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( W `  D
)  =  ( A 
\  ( ( _|_
P `  K ) `  { D } ) ) )
54eleq2d 2350 . 2  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( P  e.  ( W `  D )  <-> 
P  e.  ( A 
\  ( ( _|_
P `  K ) `  { D } ) ) ) )
6 eldif 3162 . 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 1623    e. wcel 1684    \ cdif 3149   {csn 3640   ` cfv 5255   Atomscatm 29453   _|_
PcpolN 30091   WAtomscwpointsN 30175
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-watsN 30179
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