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Mirrors > Home > ILE Home > Th. List > dedekindicclemloc | Unicode version |
Description: Lemma for dedekindicc 13053. The set L is located. (Contributed by Jim Kingdon, 15-Feb-2024.) |
Ref | Expression |
---|---|
dedekindicc.a | |
dedekindicc.b | |
dedekindicc.lss | |
dedekindicc.uss | |
dedekindicc.lm | |
dedekindicc.um | |
dedekindicc.lr | |
dedekindicc.ur | |
dedekindicc.disj | |
dedekindicc.loc |
Ref | Expression |
---|---|
dedekindicclemloc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 3970 | . . . . 5 | |
2 | eleq1w 2218 | . . . . . 6 | |
3 | 2 | orbi2d 780 | . . . . 5 |
4 | 1, 3 | imbi12d 233 | . . . 4 |
5 | breq1 3969 | . . . . . . 7 | |
6 | eleq1w 2218 | . . . . . . . 8 | |
7 | 6 | orbi1d 781 | . . . . . . 7 |
8 | 5, 7 | imbi12d 233 | . . . . . 6 |
9 | 8 | ralbidv 2457 | . . . . 5 |
10 | dedekindicc.loc | . . . . . 6 | |
11 | 10 | adantr 274 | . . . . 5 |
12 | simprl 521 | . . . . 5 | |
13 | 9, 11, 12 | rspcdva 2821 | . . . 4 |
14 | simprr 522 | . . . 4 | |
15 | 4, 13, 14 | rspcdva 2821 | . . 3 |
16 | simpr 109 | . . . . . . 7 | |
17 | 5 | rexbidv 2458 | . . . . . . . . 9 |
18 | 6, 17 | bibi12d 234 | . . . . . . . 8 |
19 | dedekindicc.lr | . . . . . . . . 9 | |
20 | 19 | ad2antrr 480 | . . . . . . . 8 |
21 | 12 | adantr 274 | . . . . . . . 8 |
22 | 18, 20, 21 | rspcdva 2821 | . . . . . . 7 |
23 | 16, 22 | mpbid 146 | . . . . . 6 |
24 | breq2 3970 | . . . . . . 7 | |
25 | 24 | cbvrexv 2681 | . . . . . 6 |
26 | 23, 25 | sylib 121 | . . . . 5 |
27 | 26 | ex 114 | . . . 4 |
28 | dedekindicc.a | . . . . . . 7 | |
29 | 28 | ad2antrr 480 | . . . . . 6 |
30 | dedekindicc.b | . . . . . . 7 | |
31 | 30 | ad2antrr 480 | . . . . . 6 |
32 | dedekindicc.lss | . . . . . . 7 | |
33 | 32 | ad2antrr 480 | . . . . . 6 |
34 | dedekindicc.uss | . . . . . . 7 | |
35 | 34 | ad2antrr 480 | . . . . . 6 |
36 | dedekindicc.lm | . . . . . . 7 | |
37 | 36 | ad2antrr 480 | . . . . . 6 |
38 | dedekindicc.um | . . . . . . 7 | |
39 | 38 | ad2antrr 480 | . . . . . 6 |
40 | 19 | ad2antrr 480 | . . . . . 6 |
41 | dedekindicc.ur | . . . . . . 7 | |
42 | 41 | ad2antrr 480 | . . . . . 6 |
43 | dedekindicc.disj | . . . . . . 7 | |
44 | 43 | ad2antrr 480 | . . . . . 6 |
45 | 10 | ad2antrr 480 | . . . . . 6 |
46 | simpr 109 | . . . . . 6 | |
47 | 29, 31, 33, 35, 37, 39, 40, 42, 44, 45, 46 | dedekindicclemuub 13046 | . . . . 5 |
48 | 47 | ex 114 | . . . 4 |
49 | 27, 48 | orim12d 776 | . . 3 |
50 | 15, 49 | syld 45 | . 2 |
51 | 50 | ralrimivva 2539 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 wceq 1335 wcel 2128 wral 2435 wrex 2436 cin 3101 wss 3102 c0 3394 class class class wbr 3966 (class class class)co 5825 cr 7732 clt 7913 cicc 9796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-id 4254 df-po 4257 df-iso 4258 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-iota 5136 df-fun 5173 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-icc 9800 |
This theorem is referenced by: dedekindicclemlub 13049 |
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