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Theorem elat2 23796
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 23795 . 2  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  0H  <oH  A ) )
2 h0elch 22710 . . . . 5  |-  0H  e.  CH
3 cvbr2 23739 . . . . 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 22900 . . . . 5  |-  ( A  e.  CH  ->  ( 0H  C.  A  <->  A  =/=  0H ) )
6 ch0pss 22900 . . . . . . . . . 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 2651 . . . . . . . . . 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 2698 . . . . . 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 1649    e. wcel 1721    =/= wne 2567   A.wral 2666    C_ wss 3280    C. wpss 3281   class class class wbr 4172   CHcch 22385   0Hc0h 22391    <oH ccv 22420  HAtomscat 22421
This theorem is referenced by:  atne0  23801  atss  23802  h1da  23805  atom1d  23809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-icc 10879  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-lm 17247  df-haus 17333  df-grpo 21732  df-gid 21733  df-ginv 21734  df-gdiv 21735  df-ablo 21823  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-vs 22031  df-nmcv 22032  df-ims 22033  df-hnorm 22424  df-hvsub 22427  df-hlim 22428  df-sh 22662  df-ch 22677  df-ch0 22708  df-cv 23735  df-at 23794
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