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Mirrors > Home > ILE Home > Th. List > ivthinclemloc | Unicode version |
Description: Lemma for ivthinc 13415. 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 3993 | . . . . . . . 8 | |
3 | fveq2 5496 | . . . . . . . . 9 | |
4 | 3 | breq2d 4001 | . . . . . . . 8 |
5 | 2, 4 | imbi12d 233 | . . . . . . 7 |
6 | breq1 3992 | . . . . . . . . . 10 | |
7 | fveq2 5496 | . . . . . . . . . . 11 | |
8 | 7 | breq1d 3999 | . . . . . . . . . 10 |
9 | 6, 8 | imbi12d 233 | . . . . . . . . 9 |
10 | 9 | ralbidv 2470 | . . . . . . . 8 |
11 | ivthinc.i | . . . . . . . . . . . 12 | |
12 | 11 | expr 373 | . . . . . . . . . . 11 |
13 | 12 | ralrimiva 2543 | . . . . . . . . . 10 |
14 | 13 | ralrimiva 2543 | . . . . . . . . 9 |
15 | 14 | ad2antrr 485 | . . . . . . . 8 |
16 | simplrl 530 | . . . . . . . 8 | |
17 | 10, 15, 16 | rspcdva 2839 | . . . . . . 7 |
18 | simplrr 531 | . . . . . . 7 | |
19 | 5, 17, 18 | rspcdva 2839 | . . . . . 6 |
20 | 1, 19 | mpd 13 | . . . . 5 |
21 | 7 | eleq1d 2239 | . . . . . . 7 |
22 | ivth.8 | . . . . . . . . 9 | |
23 | 22 | ralrimiva 2543 | . . . . . . . 8 |
24 | 23 | ad2antrr 485 | . . . . . . 7 |
25 | 21, 24, 16 | rspcdva 2839 | . . . . . 6 |
26 | fveq2 5496 | . . . . . . . 8 | |
27 | 26 | eleq1d 2239 | . . . . . . 7 |
28 | 27, 24, 18 | rspcdva 2839 | . . . . . 6 |
29 | ivth.3 | . . . . . . 7 | |
30 | 29 | ad2antrr 485 | . . . . . 6 |
31 | axltwlin 7987 | . . . . . 6 | |
32 | 25, 28, 30, 31 | syl3anc 1233 | . . . . 5 |
33 | 20, 32 | mpd 13 | . . . 4 |
34 | 16 | adantr 274 | . . . . . . 7 |
35 | simpr 109 | . . . . . . 7 | |
36 | fveq2 5496 | . . . . . . . . 9 | |
37 | 36 | breq1d 3999 | . . . . . . . 8 |
38 | ivthinclem.l | . . . . . . . 8 | |
39 | 37, 38 | elrab2 2889 | . . . . . . 7 |
40 | 34, 35, 39 | sylanbrc 415 | . . . . . 6 |
41 | 40 | ex 114 | . . . . 5 |
42 | 18 | adantr 274 | . . . . . . 7 |
43 | simpr 109 | . . . . . . 7 | |
44 | fveq2 5496 | . . . . . . . . 9 | |
45 | 44 | breq2d 4001 | . . . . . . . 8 |
46 | ivthinclem.r | . . . . . . . 8 | |
47 | 45, 46 | elrab2 2889 | . . . . . . 7 |
48 | 42, 43, 47 | sylanbrc 415 | . . . . . 6 |
49 | 48 | ex 114 | . . . . 5 |
50 | 41, 49 | orim12d 781 | . . . 4 |
51 | 33, 50 | mpd 13 | . . 3 |
52 | 51 | ex 114 | . 2 |
53 | 52 | ralrimivva 2552 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 703 wceq 1348 wcel 2141 wral 2448 crab 2452 wss 3121 class class class wbr 3989 cfv 5198 (class class class)co 5853 cc 7772 cr 7773 clt 7954 cicc 9848 ccncf 13351 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltwlin 7887 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-iota 5160 df-fv 5206 df-pnf 7956 df-mnf 7957 df-ltxr 7959 |
This theorem is referenced by: ivthinclemex 13414 |
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