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Mirrors > Home > ILE Home > Th. List > ivthinclemloc | Unicode version |
Description: Lemma for ivthinc 12790. 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 3933 | . . . . . . . 8 | |
3 | fveq2 5421 | . . . . . . . . 9 | |
4 | 3 | breq2d 3941 | . . . . . . . 8 |
5 | 2, 4 | imbi12d 233 | . . . . . . 7 |
6 | breq1 3932 | . . . . . . . . . 10 | |
7 | fveq2 5421 | . . . . . . . . . . 11 | |
8 | 7 | breq1d 3939 | . . . . . . . . . 10 |
9 | 6, 8 | imbi12d 233 | . . . . . . . . 9 |
10 | 9 | ralbidv 2437 | . . . . . . . 8 |
11 | ivthinc.i | . . . . . . . . . . . 12 | |
12 | 11 | expr 372 | . . . . . . . . . . 11 |
13 | 12 | ralrimiva 2505 | . . . . . . . . . 10 |
14 | 13 | ralrimiva 2505 | . . . . . . . . 9 |
15 | 14 | ad2antrr 479 | . . . . . . . 8 |
16 | simplrl 524 | . . . . . . . 8 | |
17 | 10, 15, 16 | rspcdva 2794 | . . . . . . 7 |
18 | simplrr 525 | . . . . . . 7 | |
19 | 5, 17, 18 | rspcdva 2794 | . . . . . 6 |
20 | 1, 19 | mpd 13 | . . . . 5 |
21 | 7 | eleq1d 2208 | . . . . . . 7 |
22 | ivth.8 | . . . . . . . . 9 | |
23 | 22 | ralrimiva 2505 | . . . . . . . 8 |
24 | 23 | ad2antrr 479 | . . . . . . 7 |
25 | 21, 24, 16 | rspcdva 2794 | . . . . . 6 |
26 | fveq2 5421 | . . . . . . . 8 | |
27 | 26 | eleq1d 2208 | . . . . . . 7 |
28 | 27, 24, 18 | rspcdva 2794 | . . . . . 6 |
29 | ivth.3 | . . . . . . 7 | |
30 | 29 | ad2antrr 479 | . . . . . 6 |
31 | axltwlin 7832 | . . . . . 6 | |
32 | 25, 28, 30, 31 | syl3anc 1216 | . . . . 5 |
33 | 20, 32 | mpd 13 | . . . 4 |
34 | 16 | adantr 274 | . . . . . . 7 |
35 | simpr 109 | . . . . . . 7 | |
36 | fveq2 5421 | . . . . . . . . 9 | |
37 | 36 | breq1d 3939 | . . . . . . . 8 |
38 | ivthinclem.l | . . . . . . . 8 | |
39 | 37, 38 | elrab2 2843 | . . . . . . 7 |
40 | 34, 35, 39 | sylanbrc 413 | . . . . . 6 |
41 | 40 | ex 114 | . . . . 5 |
42 | 18 | adantr 274 | . . . . . . 7 |
43 | simpr 109 | . . . . . . 7 | |
44 | fveq2 5421 | . . . . . . . . 9 | |
45 | 44 | breq2d 3941 | . . . . . . . 8 |
46 | ivthinclem.r | . . . . . . . 8 | |
47 | 45, 46 | elrab2 2843 | . . . . . . 7 |
48 | 42, 43, 47 | sylanbrc 413 | . . . . . 6 |
49 | 48 | ex 114 | . . . . 5 |
50 | 41, 49 | orim12d 775 | . . . 4 |
51 | 33, 50 | mpd 13 | . . 3 |
52 | 51 | ex 114 | . 2 |
53 | 52 | ralrimivva 2514 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 wceq 1331 wcel 1480 wral 2416 crab 2420 wss 3071 class class class wbr 3929 cfv 5123 (class class class)co 5774 cc 7618 cr 7619 clt 7800 cicc 9674 ccncf 12726 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltwlin 7733 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-iota 5088 df-fv 5131 df-pnf 7802 df-mnf 7803 df-ltxr 7805 |
This theorem is referenced by: ivthinclemex 12789 |
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