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Theorem cdleme0nex 29168
 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 29089- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 28222, 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 989 . . . 4
2 simp12 991 . . . 4
31, 2jca 520 . . 3
4 simp3l 988 . . . . . 6
5 simp13 992 . . . . . . 7
6 ralnex 2517 . . . . . . 7
75, 6sylibr 205 . . . . . 6
8 breq1 3923 . . . . . . . . . 10
98notbid 287 . . . . . . . . 9
10 oveq2 5718 . . . . . . . . . 10
11 oveq2 5718 . . . . . . . . . 10
1210, 11eqeq12d 2267 . . . . . . . . 9
139, 12anbi12d 694 . . . . . . . 8
1413notbid 287 . . . . . . 7
1514rcla4va 2819 . . . . . 6
164, 7, 15syl2anc 645 . . . . 5
17 simp11 990 . . . . . . . 8
18 hlcvl 28238 . . . . . . . 8
1917, 18syl 17 . . . . . . 7
20 simp21 993 . . . . . . 7
21 simp22 994 . . . . . . 7
22 simp23 995 . . . . . . 7
23 cdleme0nex.a . . . . . . . 8
24 cdleme0nex.l . . . . . . . 8
25 cdleme0nex.j . . . . . . . 8
2623, 24, 25cvlsupr2 28222 . . . . . . 7
2719, 20, 21, 4, 22, 26syl131anc 1200 . . . . . 6
2827anbi2d 687 . . . . 5
2916, 28mtbid 293 . . . 4
30 ianor 476 . . . . 5
31 df-3an 941 . . . . . . . 8
3231anbi2i 678 . . . . . . 7
33 an12 775 . . . . . . 7
3432, 33bitri 242 . . . . . 6
3534notbii 289 . . . . 5
36 pm4.62 410 . . . . 5
3730, 35, 363bitr4ri 271 . . . 4
3829, 37sylibr 205 . . 3
393, 38mt2d 111 . 2
40 neanior 2497 . . 3
4140con2bii 324 . 2
4239, 41sylibr 205 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wb 178   wo 359   wa 360   w3a 939   wceq 1619   wcel 1621   wne 2412  wral 2509  wrex 2510   class class class wbr 3920  cfv 4592  (class class class)co 5710  cple 13089  cjn 13922  catm 28142  clc 28144  chlt 28229 This theorem is referenced by:  cdleme18c  29171  cdleme18d  29173  cdlemg17b  29540  cdlemg17h  29546 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-join 13954  df-lat 13996  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230
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