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Theorem 4atexlem7 33565
 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 32834, our is a shorter way to express . With a longer proof, the condition could be eliminated (see 4atex 33566), 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 1117 . . . . . 6
2 simp1r1 1102 . . . . . . 7
323ad2ant1 1027 . . . . . 6
4 simp1r2 1103 . . . . . . 7
543ad2ant1 1027 . . . . . 6
6 simp2 1007 . . . . . . 7
7 simp3l 1034 . . . . . . 7
86, 7jca 535 . . . . . 6
9 simp1r3 1104 . . . . . . 7
1093ad2ant1 1027 . . . . . 6
11 simp3r 1035 . . . . . 6
12 simp12 1037 . . . . . 6
13 simp13 1038 . . . . . 6
14 4that.l . . . . . . 7
15 4that.j . . . . . . 7
16 eqid 2423 . . . . . . 7
17 4that.a . . . . . . 7
18 4that.h . . . . . . 7
1914, 15, 16, 17, 184atexlemex6 33564 . . . . . 6
201, 3, 5, 8, 10, 11, 12, 13, 19syl323anc 1295 . . . . 5
2120rexlimdv3a 2920 . . . 4
22213exp 1205 . . 3
23223impd 1220 . 2
24233impia 1203 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 371   w3a 983   wceq 1438   wcel 1869   wne 2619  wrex 2777   class class class wbr 4421  cfv 5599  (class class class)co 6303  cple 15190  cjn 16182  cmee 16183  catm 32754  chlt 32841  clh 33474 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-p1 16279  df-lat 16285  df-clat 16347  df-oposet 32667  df-ol 32669  df-oml 32670  df-covers 32757  df-ats 32758  df-atl 32789  df-cvlat 32813  df-hlat 32842  df-llines 32988  df-lplanes 32989  df-lhyp 33478 This theorem is referenced by:  4atex  33566  cdleme21i  33827
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