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Theorem cdleme0nex 31024
 Description: Part of proof of Lemma E in [Crawley] p. 114, 4th line of 4th paragraph. Whenever (in their terminology) p q/0 (i.e. the sublattice from 0 to p q) contains precisely three atoms, any atom not under w must equal either p or q. (In case of 3 atoms, one of them must be u - see cdleme0a 30945- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 30078, our is a shorter way to express . Thus, the negated existential condition states there are no atoms different from p or q that are also not under w. (Contributed by NM, 12-Nov-2012.)
Hypotheses
Ref Expression
cdleme0nex.l
cdleme0nex.j
cdleme0nex.a
Assertion
Ref Expression
cdleme0nex
Distinct variable groups:   ,   ,   ,   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem cdleme0nex
StepHypRef Expression
1 simp3r 986 . . . 4
2 simp12 988 . . . 4
31, 2jca 519 . . 3
4 simp3l 985 . . . . . 6
5 simp13 989 . . . . . . 7
6 ralnex 2707 . . . . . . 7
75, 6sylibr 204 . . . . . 6
8 breq1 4207 . . . . . . . . . 10
98notbid 286 . . . . . . . . 9
10 oveq2 6081 . . . . . . . . . 10
11 oveq2 6081 . . . . . . . . . 10
1210, 11eqeq12d 2449 . . . . . . . . 9
139, 12anbi12d 692 . . . . . . . 8
1413notbid 286 . . . . . . 7
1514rspcva 3042 . . . . . 6
164, 7, 15syl2anc 643 . . . . 5
17 simp11 987 . . . . . . . 8
18 hlcvl 30094 . . . . . . . 8
1917, 18syl 16 . . . . . . 7
20 simp21 990 . . . . . . 7
21 simp22 991 . . . . . . 7
22 simp23 992 . . . . . . 7
23 cdleme0nex.a . . . . . . . 8
24 cdleme0nex.l . . . . . . . 8
25 cdleme0nex.j . . . . . . . 8
2623, 24, 25cvlsupr2 30078 . . . . . . 7
2719, 20, 21, 4, 22, 26syl131anc 1197 . . . . . 6
2827anbi2d 685 . . . . 5
2916, 28mtbid 292 . . . 4
30 ianor 475 . . . . 5
31 df-3an 938 . . . . . . . 8
3231anbi2i 676 . . . . . . 7
33 an12 773 . . . . . . 7
3432, 33bitri 241 . . . . . 6
3534notbii 288 . . . . 5
36 pm4.62 409 . . . . 5
3730, 35, 363bitr4ri 270 . . . 4
3829, 37sylibr 204 . . 3
393, 38mt2d 111 . 2
40 neanior 2683 . . 3
4140con2bii 323 . 2
4239, 41sylibr 204 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wo 358   wa 359   w3a 936   wceq 1652   wcel 1725   wne 2598  wral 2697  wrex 2698   class class class wbr 4204  cfv 5446  (class class class)co 6073  cple 13528  cjn 14393  catm 29998  clc 30000  chlt 30085 This theorem is referenced by:  cdleme18c  31027  cdleme18d  31029  cdlemg17b  31396  cdlemg17h  31402 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 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-nel 2601  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-pw 3793  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-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-join 14425  df-lat 14467  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086
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