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Theorem elat2 22920
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 a 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 22919 . 2  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  0H  <oH  A ) )
2 h0elch 21834 . . . . 5  |-  0H  e.  CH
3 cvbr2 22863 . . . . 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 651 . . . 4  |-  ( A  e.  CH  ->  ( 0H  <oH  A  <->  ( 0H  C.  A  /\  A. x  e.  CH  ( ( 0H 
C.  x  /\  x  C_  A )  ->  x  =  A ) ) ) )
5 ch0pss 22024 . . . . 5  |-  ( A  e.  CH  ->  ( 0H  C.  A  <->  A  =/=  0H ) )
6 ch0pss 22024 . . . . . . . . . 10  |-  ( x  e.  CH  ->  ( 0H  C.  x  <->  x  =/=  0H ) )
76imbi1d 308 . . . . . . . . 9  |-  ( x  e.  CH  ->  (
( 0H  C.  x  ->  x  =  A )  <-> 
( x  =/=  0H  ->  x  =  A ) ) )
87imbi2d 307 . . . . . . . 8  |-  ( x  e.  CH  ->  (
( x  C_  A  ->  ( 0H  C.  x  ->  x  =  A ) )  <->  ( x  C_  A  ->  ( x  =/= 
0H  ->  x  =  A ) ) ) )
9 impexp 433 . . . . . . . . 9  |-  ( ( ( 0H  C.  x  /\  x  C_  A )  ->  x  =  A )  <->  ( 0H  C.  x  ->  ( x  C_  A  ->  x  =  A ) ) )
10 bi2.04 350 . . . . . . . . 9  |-  ( ( 0H  C.  x  -> 
( x  C_  A  ->  x  =  A ) )  <->  ( x  C_  A  ->  ( 0H  C.  x  ->  x  =  A ) ) )
119, 10bitri 240 . . . . . . . 8  |-  ( ( ( 0H  C.  x  /\  x  C_  A )  ->  x  =  A )  <->  ( x  C_  A  ->  ( 0H  C.  x  ->  x  =  A ) ) )
12 orcom 376 . . . . . . . . . 10  |-  ( ( x  =  A  \/  x  =  0H )  <->  ( x  =  0H  \/  x  =  A )
)
13 neor 2530 . . . . . . . . . 10  |-  ( ( x  =  0H  \/  x  =  A )  <->  ( x  =/=  0H  ->  x  =  A ) )
1412, 13bitri 240 . . . . . . . . 9  |-  ( ( x  =  A  \/  x  =  0H )  <->  ( x  =/=  0H  ->  x  =  A ) )
1514imbi2i 303 . . . . . . . 8  |-  ( ( x  C_  A  ->  ( x  =  A  \/  x  =  0H )
)  <->  ( x  C_  A  ->  ( x  =/= 
0H  ->  x  =  A ) ) )
168, 11, 153bitr4g 279 . . . . . . 7  |-  ( x  e.  CH  ->  (
( ( 0H  C.  x  /\  x  C_  A
)  ->  x  =  A )  <->  ( x  C_  A  ->  ( x  =  A  \/  x  =  0H ) ) ) )
1716ralbiia 2575 . . . . . 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 10 . . . . 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 691 . . . 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 245 . . 3  |-  ( A  e.  CH  ->  (
( A  =/=  0H  /\ 
A. x  e.  CH  ( x  C_  A  -> 
( x  =  A  \/  x  =  0H ) ) )  <->  0H  <oH  A ) )
2120pm5.32i 618 . 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 243 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 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152    C. wpss 3153   class class class wbr 4023   CHcch 21509   0Hc0h 21515    <oH ccv 21544  HAtomscat 21545
This theorem is referenced by:  atne0  22925  atss  22926  h1da  22929  atom1d  22933
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-lm 16959  df-haus 17043  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-hnorm 21548  df-hvsub 21551  df-hlim 21552  df-sh 21786  df-ch 21801  df-ch0 21832  df-cv 22859  df-at 22918
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