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Theorem dedekindicclemicc 15443

Description: Lemma for dedekindicc 15444. Same as dedekindicc 15444, except that we merely show

to be an element of

. Later we will strengthen that to

. (Contributed by Jim Kingdon, 5-Jan-2024.)

**Hypotheses**
Ref	Expression
dedekindicc.a
dedekindicc.b
dedekindicc.lss
dedekindicc.uss
dedekindicc.lm
dedekindicc.um
dedekindicc.lr
dedekindicc.ur
dedekindicc.disj
dedekindicc.loc	$\/$
dedekindicc.ab

**Assertion**
Ref	Expression
dedekindicclemicc	$/\$

Distinct variable groups:

Proof of Theorem dedekindicclemicc Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dedekindicc.a	. . 3
2		dedekindicc.b	. . 3
3		dedekindicc.lss	. . 3
4		dedekindicc.uss	. . 3
5		dedekindicc.lm	. . 3
6		dedekindicc.um	. . 3
7		dedekindicc.lr	. . 3
8		dedekindicc.ur	. . 3
9		dedekindicc.disj	. . 3
10		dedekindicc.loc	. . 3 $\/$
11		dedekindicc.ab	. . 3
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	dedekindicclemlu 15441	. 2 $/\$
13	1	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
14	2	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
15	3	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
16	4	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
17	5	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
18	6	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
19	7	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
20	8	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
21	9	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
22	10	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$ $\/$
23	11	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
24		simprll 539	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
25	24	ad3antrrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
26		simprr 533	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
27	26	ad3antrrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$ $/\$
28		simprlr 540	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
29	28	ad3antrrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
30		simpllr 536	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$ $/\$
31		simpr 110	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
32	13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 27, 29, 30, 31	dedekindicclemeu 15442	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
33	1	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
34	2	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
35	3	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
36	4	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
37	5	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
38	6	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
39	7	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
40	8	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
41	9	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
42	10	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$ $\/$
43	11	ad4antr 494	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
44	28	ad3antrrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
45		simpllr 536	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$ $/\$
46	24	ad3antrrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
47	26	ad3antrrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$ $/\$
48		simpr 110	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
49	33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48	dedekindicclemeu 15442	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
50		simpr 110	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
51		iccssre 10251	. . . . . . . . . . . . . 14 $/\$
52	1, 2, 51	syl2anc 411	. . . . . . . . . . . . 13
53	52	ad3antrrr 492	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
54	24	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
55	53, 54	sseldd 3229	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
56	28	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
57	53, 56	sseldd 3229	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
58		reaplt 8827	. . . . . . . . . . 11 $/\$ # $\/$
59	55, 57, 58	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$
60	50, 59	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $\/$
61	32, 49, 60	mpjaodan 806	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
62	61	inegd 1417	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
63	52	ad2antrr 488	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	24	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	63, 64	sseldd 3229	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
66	65	recnd 8267	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	28	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
68	63, 67	sseldd 3229	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69	68	recnd 8267	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70		apti 8861	. . . . . . . 8 $/\$ #
71	66, 69, 70	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
72	62, 71	mpbird 167	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		ancom 266	. . . . . . . . . . . . . . 15 $/\$ $/\$
74	73	anbi2i 457	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75		anass 401	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
76	74, 75	bitr4i 187	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
77	76	anbi2i 457	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78		anass 401	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
79	77, 78	bitr4i 187	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
80	79	anbi1i 458	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
81		anass 401	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
82	80, 81	bitri 184	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
83		anass 401	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
84	83	bicomi 132	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
85	84	anbi1i 458	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
86		anass 401	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
87	85, 86	bitri 184	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
88	82, 87	bitri 184	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
89		anass 401	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
90		ancom 266	. . . . . . . . . . 11 $/\$ $/\$
91	90	anbi1i 458	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
92	89, 91	bitr3i 186	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
93	92	anbi2i 457	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
94	88, 93	bitri 184	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
95	94	imbi1i 238	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
96	72, 95	mpbi 145	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
97	96	ex 115	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
98	97	ralrimivva 2615	. . 3 $/\$ $/\$ $/\$
99		breq2 4097	. . . . . 6
100	99	ralbidv 2533	. . . . 5
101		breq1 4096	. . . . . 6
102	101	ralbidv 2533	. . . . 5
103	100, 102	anbi12d 473	. . . 4 $/\$ $/\$
104	103	rmo4 3000	. . 3 $/\$ $/\$ $/\$ $/\$
105	98, 104	sylibr 134	. 2 $/\$
106		reu5 2752	. 2 $/\$ $/\$ $/\$ $/\$
107	12, 105, 106	sylanbrc 417	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716

wceq 1398

wfal 1403

wcel 2202

wral 2511

wrex 2512

wreu 2513

wrmo 2514

cin 3200

wss 3201

c0 3496 class class class wbr 4093 (class class class)co 6028

cc 8090

cr 8091

clt 8273 # cap 8820

cicc 10187

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-precex 8202 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-ltwlin 8205 ax-pre-lttrn 8206 ax-pre-apti 8207 ax-pre-ltadd 8208 ax-pre-mulgt0 8209 ax-pre-mulext 8210 ax-arch 8211 ax-caucvg 8212 ax-pre-suploc 8213

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-sup 7243 df-inf 7244 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278 df-le 8279 df-sub 8411 df-neg 8412 df-reap 8814 df-ap 8821 df-div 8912 df-inn 9203 df-2 9261 df-3 9262 df-4 9263 df-n0 9462 df-z 9541 df-uz 9817 df-rp 9950 df-icc 10191 df-seqfrec 10773 df-exp 10864 df-cj 11482 df-re 11483 df-im 11484 df-rsqrt 11638 df-abs 11639

This theorem is referenced by: dedekindicc 15444

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