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Mirrors > Home > ILE Home > Th. List > caucvgprprlemell | Unicode version |
Description: Lemma for caucvgprpr 6810. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.) |
Ref | Expression |
---|---|
caucvgprprlemell.lim |
Ref | Expression |
---|---|
caucvgprprlemell |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5519 | . . . . . . . 8 | |
2 | 1 | breq2d 3776 | . . . . . . 7 |
3 | 2 | abbidv 2155 | . . . . . 6 |
4 | 1 | breq1d 3774 | . . . . . . 7 |
5 | 4 | abbidv 2155 | . . . . . 6 |
6 | 3, 5 | opeq12d 3557 | . . . . 5 |
7 | 6 | breq1d 3774 | . . . 4 |
8 | 7 | rexbidv 2327 | . . 3 |
9 | caucvgprprlemell.lim | . . . . 5 | |
10 | 9 | fveq2i 5181 | . . . 4 |
11 | nqex 6461 | . . . . . 6 | |
12 | 11 | rabex 3901 | . . . . 5 |
13 | 11 | rabex 3901 | . . . . 5 |
14 | 12, 13 | op1st 5773 | . . . 4 |
15 | 10, 14 | eqtri 2060 | . . 3 |
16 | 8, 15 | elrab2 2700 | . 2 |
17 | opeq1 3549 | . . . . . . . . . . . 12 | |
18 | 17 | eceq1d 6142 | . . . . . . . . . . 11 |
19 | 18 | fveq2d 5182 | . . . . . . . . . 10 |
20 | 19 | oveq2d 5528 | . . . . . . . . 9 |
21 | 20 | breq2d 3776 | . . . . . . . 8 |
22 | 21 | abbidv 2155 | . . . . . . 7 |
23 | 20 | breq1d 3774 | . . . . . . . 8 |
24 | 23 | abbidv 2155 | . . . . . . 7 |
25 | 22, 24 | opeq12d 3557 | . . . . . 6 |
26 | fveq2 5178 | . . . . . 6 | |
27 | 25, 26 | breq12d 3777 | . . . . 5 |
28 | 27 | cbvrexv 2534 | . . . 4 |
29 | opeq1 3549 | . . . . . . . . . . . 12 | |
30 | 29 | eceq1d 6142 | . . . . . . . . . . 11 |
31 | 30 | fveq2d 5182 | . . . . . . . . . 10 |
32 | 31 | oveq2d 5528 | . . . . . . . . 9 |
33 | 32 | breq2d 3776 | . . . . . . . 8 |
34 | 33 | abbidv 2155 | . . . . . . 7 |
35 | 32 | breq1d 3774 | . . . . . . . 8 |
36 | 35 | abbidv 2155 | . . . . . . 7 |
37 | 34, 36 | opeq12d 3557 | . . . . . 6 |
38 | fveq2 5178 | . . . . . 6 | |
39 | 37, 38 | breq12d 3777 | . . . . 5 |
40 | 39 | cbvrexv 2534 | . . . 4 |
41 | 28, 40 | bitri 173 | . . 3 |
42 | 41 | anbi2i 430 | . 2 |
43 | 16, 42 | bitri 173 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wcel 1393 cab 2026 wrex 2307 crab 2310 cop 3378 class class class wbr 3764 cfv 4902 (class class class)co 5512 c1st 5765 c1o 5994 cec 6104 cnpi 6370 ceq 6377 cnq 6378 cplq 6380 crq 6382 cltq 6383 cpp 6391 cltp 6393 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-1st 5767 df-ec 6108 df-qs 6112 df-ni 6402 df-nqqs 6446 |
This theorem is referenced by: caucvgprprlemopl 6795 caucvgprprlemlol 6796 caucvgprprlemdisj 6800 caucvgprprlemloc 6801 |
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