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Theorem dedekindicclemicc 15306

Description: Lemma for dedekindicc 15307. Same as dedekindicc 15307, 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 15304	. 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 537	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
25	24	ad3antrrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
26		simprr 531	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
27	26	ad3antrrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$ $/\$
28		simprlr 538	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
29	28	ad3antrrr 492	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
30		simpllr 534	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$ $/\$
31		simpr 110	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
32	13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 27, 29, 30, 31	dedekindicclemeu 15305	. . . . . . . . 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 534	. . . . . . . . . 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 15305	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $/\$
50		simpr 110	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
51		iccssre 10151	. . . . . . . . . . . . . 14 $/\$
52	1, 2, 51	syl2anc 411	. . . . . . . . . . . . 13
53	52	ad3antrrr 492	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
54	24	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
55	53, 54	sseldd 3225	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
56	28	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
57	53, 56	sseldd 3225	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
58		reaplt 8735	. . . . . . . . . . 11 $/\$ # $\/$
59	55, 57, 58	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$
60	50, 59	mpbid 147	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $\/$
61	32, 49, 60	mpjaodan 803	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
62	61	inegd 1414	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
63	52	ad2antrr 488	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	24	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	63, 64	sseldd 3225	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
66	65	recnd 8175	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	28	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
68	63, 67	sseldd 3225	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69	68	recnd 8175	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70		apti 8769	. . . . . . . 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 2612	. . 3 $/\$ $/\$ $/\$
99		breq2 4087	. . . . . 6
100	99	ralbidv 2530	. . . . 5
101		breq1 4086	. . . . . 6
102	101	ralbidv 2530	. . . . 5
103	100, 102	anbi12d 473	. . . 4 $/\$ $/\$
104	103	rmo4 2996	. . 3 $/\$ $/\$ $/\$ $/\$
105	98, 104	sylibr 134	. 2 $/\$
106		reu5 2749	. 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 713

wceq 1395

wfal 1400

wcel 2200

wral 2508

wrex 2509

wreu 2510

wrmo 2511

cin 3196

wss 3197

c0 3491 class class class wbr 4083 (class class class)co 6001

cc 7997

cr 7998

clt 8181 # cap 8728

cicc 10087

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-mulext 8117 ax-arch 8118 ax-caucvg 8119 ax-pre-suploc 8120

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-recs 6451 df-frec 6537 df-sup 7151 df-inf 7152 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-div 8820 df-inn 9111 df-2 9169 df-3 9170 df-4 9171 df-n0 9370 df-z 9447 df-uz 9723 df-rp 9850 df-icc 10091 df-seqfrec 10670 df-exp 10761 df-cj 11353 df-re 11354 df-im 11355 df-rsqrt 11509 df-abs 11510

This theorem is referenced by: dedekindicc 15307

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