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Theorem 4atexlem7 30946
 Description: Whenever there are at least 4 atoms under (specifically, , , , and ), there are also at least 4 atoms under . This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p q/0 and hence p s/0 contains at least four atoms..." Note that by cvlsupr2 30215, our is a shorter way to express . With a longer proof, the condition could be eliminated (see 4atex 30947), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.)
Hypotheses
Ref Expression
4that.l
4that.j
4that.a
4that.h
Assertion
Ref Expression
4atexlem7
Distinct variable groups:   ,,   ,   ,,   ,,   ,,   ,,   ,,   ,,   ,,
Allowed substitution hint:   ()

Proof of Theorem 4atexlem7
StepHypRef Expression
1 simp11l 1069 . . . . . 6
2 simp1r1 1054 . . . . . . 7
323ad2ant1 979 . . . . . 6
4 simp1r2 1055 . . . . . . 7
543ad2ant1 979 . . . . . 6
6 simp2 959 . . . . . . 7
7 simp3l 986 . . . . . . 7
86, 7jca 520 . . . . . 6
9 simp1r3 1056 . . . . . . 7
1093ad2ant1 979 . . . . . 6
11 simp3r 987 . . . . . 6
12 simp12 989 . . . . . 6
13 simp13 990 . . . . . 6
14 4that.l . . . . . . 7
15 4that.j . . . . . . 7
16 eqid 2438 . . . . . . 7
17 4that.a . . . . . . 7
18 4that.h . . . . . . 7
1914, 15, 16, 17, 184atexlemex6 30945 . . . . . 6
201, 3, 5, 8, 10, 11, 12, 13, 19syl323anc 1215 . . . . 5
2120rexlimdv3a 2834 . . . 4
22213exp 1153 . . 3
23223impd 1168 . 2
24233impia 1151 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726   wne 2601  wrex 2708   class class class wbr 4215  cfv 5457  (class class class)co 6084  cple 13541  cjn 14406  cmee 14407  catm 30135  chlt 30222  clh 30855 This theorem is referenced by:  4atex  30947  cdleme21i  31206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lhyp 30859
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