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Theorem 4atexlem7 30264
 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 29533, our is a shorter way to express . With a longer proof, the condition could be eliminated (see 4atex 30265), 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 1066 . . . . . 6
2 simp1r1 1051 . . . . . . 7
323ad2ant1 976 . . . . . 6
4 simp1r2 1052 . . . . . . 7
543ad2ant1 976 . . . . . 6
6 simp2 956 . . . . . . 7
7 simp3l 983 . . . . . . 7
86, 7jca 518 . . . . . 6
9 simp1r3 1053 . . . . . . 7
1093ad2ant1 976 . . . . . 6
11 simp3r 984 . . . . . 6
12 simp12 986 . . . . . 6
13 simp13 987 . . . . . 6
14 4that.l . . . . . . 7
15 4that.j . . . . . . 7
16 eqid 2283 . . . . . . 7
17 4that.a . . . . . . 7
18 4that.h . . . . . . 7
1914, 15, 16, 17, 184atexlemex6 30263 . . . . . 6
201, 3, 5, 8, 10, 11, 12, 13, 19syl323anc 1212 . . . . 5
2120rexlimdv3a 2669 . . . 4
22213exp 1150 . . 3
23223impd 1165 . 2
24233impia 1148 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358   w3a 934   wceq 1623   wcel 1684   wne 2446  wrex 2544   class class class wbr 4023  cfv 5255  (class class class)co 5858  cple 13215  cjn 14078  cmee 14079  catm 29453  chlt 29540  clh 30173 This theorem is referenced by:  4atex  30265  cdleme21i  30524 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 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lhyp 30177
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