cdlemefs32sn1aw - Mathbox for Norm Megill

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Theorem cdlemefs32sn1aw 34416

Description: Show that

is an atom not under

when

$.\/$

. (Contributed by NM, 24-Mar-2013.)

**Hypotheses**
Ref	Expression
cdlemefs32.b
cdlemefs32.l
cdlemefs32.j	$.\/$
cdlemefs32.m	$./\$
cdlemefs32.a
cdlemefs32.h
cdlemefs32.u	$.\/$ $./\$
cdlemefs32.d	$.\/$ $./\$ $.\/$ $.\/$ $./\$
cdlemefs32.e	$.\/$ $./\$ $.\/$ $.\/$ $./\$
cdlemefs32.i	$/\$ $.\/$
cdlemefs32.n	$.\/$
cdlemefs32a1.y	$.\/$ $./\$ $.\/$ $.\/$ $./\$
cdlemefs32a1.z	$/\$ $.\/$

**Assertion**
Ref	Expression
cdlemefs32sn1aw	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$

Distinct variable groups:

$.\/$ ,

$./\$ ,

Allowed substitution hints:

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

**Proof of Theorem cdlemefs32sn1aw**
Step	Hyp	Ref	Expression
1		cdlemefs32.b	. . . 4
2		fvex 5812	. . . 4
3	1, 2	eqeltri 2538	. . 3
4		nfv 1674	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
5		cdlemefs32a1.z	. . . . . . . 8 $/\$ $.\/$
6		nfra1 2810	. . . . . . . . 9 $/\$ $.\/$
7		nfcv 2616	. . . . . . . . 9
8	6, 7	nfriota 6173	. . . . . . . 8 $/\$ $.\/$
9	5, 8	nfcxfr 2614	. . . . . . 7
10	9	nfel1 2632	. . . . . 6
11		nfcv 2616	. . . . . . . 8
12		nfcv 2616	. . . . . . . 8
13	9, 11, 12	nfbr 4447	. . . . . . 7
14	13	nfn 1840	. . . . . 6
15	10, 14	nfan 1866	. . . . 5 $/\$
16	15	a1i 11	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
17	5	a1i 11	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
18		eleq1 2526	. . . . . 6
19		breq1 4406	. . . . . . 7
20	19	notbid 294	. . . . . 6
21	18, 20	anbi12d 710	. . . . 5 $/\$ $/\$
22	21	adantl 466	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$
23		simpl1 991	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
24		simpl2r 1042	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$
25		simprl 755	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
26		simprrl 763	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
27	25, 26	jca 532	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$
28		simpl2l 1041	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
29		simpl3 993	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$
30		simprrr 764	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$
31		cdlemefs32.l	. . . . . . . 8
32		cdlemefs32.j	. . . . . . . 8 $.\/$
33		cdlemefs32.m	. . . . . . . 8 $./\$
34		cdlemefs32.a	. . . . . . . 8
35		cdlemefs32.h	. . . . . . . 8
36		cdlemefs32.u	. . . . . . . 8 $.\/$ $./\$
37		cdlemefs32.d	. . . . . . . 8 $.\/$ $./\$ $.\/$ $.\/$ $./\$
38		cdlemefs32a1.y	. . . . . . . 8 $.\/$ $./\$ $.\/$ $.\/$ $./\$
39	31, 32, 33, 34, 35, 36, 37, 38	cdleme7ga 34250	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
40	31, 32, 33, 34, 35, 36, 37, 38	cdleme7 34251	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
41	39, 40	jca 532	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$
42	23, 24, 27, 28, 29, 30, 41	syl123anc 1236	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$
43	42	ex 434	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$
44		simp1 988	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
45		simp2rl 1057	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
46		simp2rr 1058	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
47		simp2l 1014	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
48		simp3 990	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
49	1, 31, 32, 33, 34, 35, 36, 37, 38, 5	cdleme25cl 34359	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
50	44, 45, 46, 47, 48, 49	syl122anc 1228	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
51		simp11 1018	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
52		simp12 1019	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
53		simp13 1020	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
54	31, 32, 34, 35	cdlemb2 34043	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
55	51, 52, 53, 47, 54	syl121anc 1224	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
56	4, 16, 17, 22, 43, 50, 55	riotasv3d 32969	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$
57	3, 56	mpan2 671	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
58		cdlemefs32.e	. . . . . 6 $.\/$ $./\$ $.\/$ $.\/$ $./\$
59		cdlemefs32.i	. . . . . 6 $/\$ $.\/$
60		cdlemefs32.n	. . . . . 6 $.\/$
61	58, 59, 60, 38, 5	cdleme31sn1c 34390	. . . . 5 $/\$ $.\/$
62	45, 48, 61	syl2anc 661	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
63	62	eleq1d 2523	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
64	62	breq1d 4413	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
65	64	notbid 294	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
66	63, 65	anbi12d 710	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$
67	57, 66	mpbird 232	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 184 $/\$ wa 369 $/\$ w3a 965

wceq 1370

wnf 1590

wcel 1758

wne 2648

wral 2799

wrex 2800

cvv 3078

csb 3398

cif 3902 class class class wbr 4403

cfv 5529

crio 6163 (class class class)co 6203

cbs 14295

cple 14367

cjn 15236

cmee 15237

catm 33266

chlt 33353

clh 33986

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-riotaBAD 32962

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-iun 4284 df-iin 4285 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-1st 6690 df-2nd 6691 df-undef 6905 df-poset 15238 df-plt 15250 df-lub 15266 df-glb 15267 df-join 15268 df-meet 15269 df-p0 15331 df-p1 15332 df-lat 15338 df-clat 15400 df-oposet 33179 df-ol 33181 df-oml 33182 df-covers 33269 df-ats 33270 df-atl 33301 df-cvlat 33325 df-hlat 33354 df-llines 33500 df-lplanes 33501 df-lvols 33502 df-lines 33503 df-psubsp 33505 df-pmap 33506 df-padd 33798 df-lhyp 33990

This theorem is referenced by: cdlemefs32snb 34417 cdleme32sn1awN 34434 cdleme32snaw 34437

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