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Mirrors > Home > ILE Home > Th. List > ivthinclemloc | Unicode version |
Description: Lemma for ivthinc 13021. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.) |
Ref | Expression |
---|---|
ivth.1 | |
ivth.2 | |
ivth.3 | |
ivth.4 | |
ivth.5 | |
ivth.7 | |
ivth.8 | |
ivth.9 | |
ivthinc.i | |
ivthinclem.l | |
ivthinclem.r |
Ref | Expression |
---|---|
ivthinclemloc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . . 6 | |
2 | breq2 3969 | . . . . . . . 8 | |
3 | fveq2 5468 | . . . . . . . . 9 | |
4 | 3 | breq2d 3977 | . . . . . . . 8 |
5 | 2, 4 | imbi12d 233 | . . . . . . 7 |
6 | breq1 3968 | . . . . . . . . . 10 | |
7 | fveq2 5468 | . . . . . . . . . . 11 | |
8 | 7 | breq1d 3975 | . . . . . . . . . 10 |
9 | 6, 8 | imbi12d 233 | . . . . . . . . 9 |
10 | 9 | ralbidv 2457 | . . . . . . . 8 |
11 | ivthinc.i | . . . . . . . . . . . 12 | |
12 | 11 | expr 373 | . . . . . . . . . . 11 |
13 | 12 | ralrimiva 2530 | . . . . . . . . . 10 |
14 | 13 | ralrimiva 2530 | . . . . . . . . 9 |
15 | 14 | ad2antrr 480 | . . . . . . . 8 |
16 | simplrl 525 | . . . . . . . 8 | |
17 | 10, 15, 16 | rspcdva 2821 | . . . . . . 7 |
18 | simplrr 526 | . . . . . . 7 | |
19 | 5, 17, 18 | rspcdva 2821 | . . . . . 6 |
20 | 1, 19 | mpd 13 | . . . . 5 |
21 | 7 | eleq1d 2226 | . . . . . . 7 |
22 | ivth.8 | . . . . . . . . 9 | |
23 | 22 | ralrimiva 2530 | . . . . . . . 8 |
24 | 23 | ad2antrr 480 | . . . . . . 7 |
25 | 21, 24, 16 | rspcdva 2821 | . . . . . 6 |
26 | fveq2 5468 | . . . . . . . 8 | |
27 | 26 | eleq1d 2226 | . . . . . . 7 |
28 | 27, 24, 18 | rspcdva 2821 | . . . . . 6 |
29 | ivth.3 | . . . . . . 7 | |
30 | 29 | ad2antrr 480 | . . . . . 6 |
31 | axltwlin 7945 | . . . . . 6 | |
32 | 25, 28, 30, 31 | syl3anc 1220 | . . . . 5 |
33 | 20, 32 | mpd 13 | . . . 4 |
34 | 16 | adantr 274 | . . . . . . 7 |
35 | simpr 109 | . . . . . . 7 | |
36 | fveq2 5468 | . . . . . . . . 9 | |
37 | 36 | breq1d 3975 | . . . . . . . 8 |
38 | ivthinclem.l | . . . . . . . 8 | |
39 | 37, 38 | elrab2 2871 | . . . . . . 7 |
40 | 34, 35, 39 | sylanbrc 414 | . . . . . 6 |
41 | 40 | ex 114 | . . . . 5 |
42 | 18 | adantr 274 | . . . . . . 7 |
43 | simpr 109 | . . . . . . 7 | |
44 | fveq2 5468 | . . . . . . . . 9 | |
45 | 44 | breq2d 3977 | . . . . . . . 8 |
46 | ivthinclem.r | . . . . . . . 8 | |
47 | 45, 46 | elrab2 2871 | . . . . . . 7 |
48 | 42, 43, 47 | sylanbrc 414 | . . . . . 6 |
49 | 48 | ex 114 | . . . . 5 |
50 | 41, 49 | orim12d 776 | . . . 4 |
51 | 33, 50 | mpd 13 | . . 3 |
52 | 51 | ex 114 | . 2 |
53 | 52 | ralrimivva 2539 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1335 wcel 2128 wral 2435 crab 2439 wss 3102 class class class wbr 3965 cfv 5170 (class class class)co 5824 cc 7730 cr 7731 clt 7912 cicc 9795 ccncf 12957 |
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 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-pre-ltwlin 7845 |
This theorem depends on definitions: df-bi 116 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-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-xp 4592 df-iota 5135 df-fv 5178 df-pnf 7914 df-mnf 7915 df-ltxr 7917 |
This theorem is referenced by: ivthinclemex 13020 |
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