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Theorem elat2 23833
Description: Expanded membership relation for the set of atoms, i.e. the predicate "is an atom (of the Hilbert lattice)." An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is the smallest nonzero element of the lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
elat2  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  ( A  =/=  0H  /\ 
A. x  e.  CH  ( x  C_  A  -> 
( x  =  A  \/  x  =  0H ) ) ) ) )
Distinct variable group:    x, A

Proof of Theorem elat2
StepHypRef Expression
1 ela 23832 . 2  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  0H  <oH  A ) )
2 h0elch 22747 . . . . 5  |-  0H  e.  CH
3 cvbr2 23776 . . . . 5  |-  ( ( 0H  e.  CH  /\  A  e.  CH )  ->  ( 0H  <oH  A  <->  ( 0H  C.  A  /\  A. x  e.  CH  ( ( 0H 
C.  x  /\  x  C_  A )  ->  x  =  A ) ) ) )
42, 3mpan 652 . . . 4  |-  ( A  e.  CH  ->  ( 0H  <oH  A  <->  ( 0H  C.  A  /\  A. x  e.  CH  ( ( 0H 
C.  x  /\  x  C_  A )  ->  x  =  A ) ) ) )
5 ch0pss 22937 . . . . 5  |-  ( A  e.  CH  ->  ( 0H  C.  A  <->  A  =/=  0H ) )
6 ch0pss 22937 . . . . . . . . . 10  |-  ( x  e.  CH  ->  ( 0H  C.  x  <->  x  =/=  0H ) )
76imbi1d 309 . . . . . . . . 9  |-  ( x  e.  CH  ->  (
( 0H  C.  x  ->  x  =  A )  <-> 
( x  =/=  0H  ->  x  =  A ) ) )
87imbi2d 308 . . . . . . . 8  |-  ( x  e.  CH  ->  (
( x  C_  A  ->  ( 0H  C.  x  ->  x  =  A ) )  <->  ( x  C_  A  ->  ( x  =/= 
0H  ->  x  =  A ) ) ) )
9 impexp 434 . . . . . . . . 9  |-  ( ( ( 0H  C.  x  /\  x  C_  A )  ->  x  =  A )  <->  ( 0H  C.  x  ->  ( x  C_  A  ->  x  =  A ) ) )
10 bi2.04 351 . . . . . . . . 9  |-  ( ( 0H  C.  x  -> 
( x  C_  A  ->  x  =  A ) )  <->  ( x  C_  A  ->  ( 0H  C.  x  ->  x  =  A ) ) )
119, 10bitri 241 . . . . . . . 8  |-  ( ( ( 0H  C.  x  /\  x  C_  A )  ->  x  =  A )  <->  ( x  C_  A  ->  ( 0H  C.  x  ->  x  =  A ) ) )
12 orcom 377 . . . . . . . . . 10  |-  ( ( x  =  A  \/  x  =  0H )  <->  ( x  =  0H  \/  x  =  A )
)
13 neor 2682 . . . . . . . . . 10  |-  ( ( x  =  0H  \/  x  =  A )  <->  ( x  =/=  0H  ->  x  =  A ) )
1412, 13bitri 241 . . . . . . . . 9  |-  ( ( x  =  A  \/  x  =  0H )  <->  ( x  =/=  0H  ->  x  =  A ) )
1514imbi2i 304 . . . . . . . 8  |-  ( ( x  C_  A  ->  ( x  =  A  \/  x  =  0H )
)  <->  ( x  C_  A  ->  ( x  =/= 
0H  ->  x  =  A ) ) )
168, 11, 153bitr4g 280 . . . . . . 7  |-  ( x  e.  CH  ->  (
( ( 0H  C.  x  /\  x  C_  A
)  ->  x  =  A )  <->  ( x  C_  A  ->  ( x  =  A  \/  x  =  0H ) ) ) )
1716ralbiia 2729 . . . . . 6  |-  ( A. x  e.  CH  ( ( 0H  C.  x  /\  x  C_  A )  ->  x  =  A )  <->  A. x  e.  CH  (
x  C_  A  ->  ( x  =  A  \/  x  =  0H )
) )
1817a1i 11 . . . . 5  |-  ( A  e.  CH  ->  ( A. x  e.  CH  (
( 0H  C.  x  /\  x  C_  A )  ->  x  =  A )  <->  A. x  e.  CH  ( x  C_  A  -> 
( x  =  A  \/  x  =  0H ) ) ) )
195, 18anbi12d 692 . . . 4  |-  ( A  e.  CH  ->  (
( 0H  C.  A  /\  A. x  e.  CH  ( ( 0H  C.  x  /\  x  C_  A
)  ->  x  =  A ) )  <->  ( A  =/=  0H  /\  A. x  e.  CH  ( x  C_  A  ->  ( x  =  A  \/  x  =  0H ) ) ) ) )
204, 19bitr2d 246 . . 3  |-  ( A  e.  CH  ->  (
( A  =/=  0H  /\ 
A. x  e.  CH  ( x  C_  A  -> 
( x  =  A  \/  x  =  0H ) ) )  <->  0H  <oH  A ) )
2120pm5.32i 619 . 2  |-  ( ( A  e.  CH  /\  ( A  =/=  0H  /\ 
A. x  e.  CH  ( x  C_  A  -> 
( x  =  A  \/  x  =  0H ) ) ) )  <-> 
( A  e.  CH  /\  0H  <oH  A ) )
221, 21bitr4i 244 1  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  ( A  =/=  0H  /\ 
A. x  e.  CH  ( x  C_  A  -> 
( x  =  A  \/  x  =  0H ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    C_ wss 3312    C. wpss 3313   class class class wbr 4204   CHcch 22422   0Hc0h 22428    <oH ccv 22457  HAtomscat 22458
This theorem is referenced by:  atne0  23838  atss  23839  h1da  23842  atom1d  23846
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-13 1727  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-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058  ax-addf 9059  ax-mulf 9060  ax-hilex 22492  ax-hfvadd 22493  ax-hvcom 22494  ax-hvass 22495  ax-hv0cl 22496  ax-hvaddid 22497  ax-hfvmul 22498  ax-hvmulid 22499  ax-hvmulass 22500  ax-hvdistr1 22501  ax-hvdistr2 22502  ax-hvmul0 22503  ax-hfi 22571  ax-his1 22574  ax-his2 22575  ax-his3 22576  ax-his4 22577
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  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-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  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-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-n0 10212  df-z 10273  df-uz 10479  df-q 10565  df-rp 10603  df-xneg 10700  df-xadd 10701  df-xmul 10702  df-icc 10913  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-topgen 13657  df-psmet 16684  df-xmet 16685  df-met 16686  df-bl 16687  df-mopn 16688  df-top 16953  df-bases 16955  df-topon 16956  df-lm 17283  df-haus 17369  df-grpo 21769  df-gid 21770  df-ginv 21771  df-gdiv 21772  df-ablo 21860  df-vc 22015  df-nv 22061  df-va 22064  df-ba 22065  df-sm 22066  df-0v 22067  df-vs 22068  df-nmcv 22069  df-ims 22070  df-hnorm 22461  df-hvsub 22464  df-hlim 22465  df-sh 22699  df-ch 22714  df-ch0 22745  df-cv 23772  df-at 23831
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