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Theorem iswatN 30728
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 30727 . . 3  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( W `  D
)  =  ( A 
\  ( ( _|_
P `  K ) `  { D } ) ) )
54eleq2d 2502 . 2  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( P  e.  ( W `  D )  <-> 
P  e.  ( A 
\  ( ( _|_
P `  K ) `  { D } ) ) ) )
6 eldif 3322 . 2  |-  ( P  e.  ( A  \ 
( ( _|_ P `  K ) `  { D } ) )  <->  ( P  e.  A  /\  -.  P  e.  ( ( _|_ P `  K ) `  { D } ) ) )
75, 6syl6bb 253 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    \ cdif 3309   {csn 3806   ` cfv 5446   Atomscatm 29998   _|_
PcpolN 30636   WAtomscwpointsN 30720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-watsN 30724
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