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Mirrors > Home > ILE Home > Th. List > caucvgprprlemell | Unicode version |
Description: Lemma for caucvgprpr 7520. 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 5781 | . . . . . . . 8 | |
2 | 1 | breq2d 3941 | . . . . . . 7 |
3 | 2 | abbidv 2257 | . . . . . 6 |
4 | 1 | breq1d 3939 | . . . . . . 7 |
5 | 4 | abbidv 2257 | . . . . . 6 |
6 | 3, 5 | opeq12d 3713 | . . . . 5 |
7 | 6 | breq1d 3939 | . . . 4 |
8 | 7 | rexbidv 2438 | . . 3 |
9 | caucvgprprlemell.lim | . . . . 5 | |
10 | 9 | fveq2i 5424 | . . . 4 |
11 | nqex 7171 | . . . . . 6 | |
12 | 11 | rabex 4072 | . . . . 5 |
13 | 11 | rabex 4072 | . . . . 5 |
14 | 12, 13 | op1st 6044 | . . . 4 |
15 | 10, 14 | eqtri 2160 | . . 3 |
16 | 8, 15 | elrab2 2843 | . 2 |
17 | opeq1 3705 | . . . . . . . . . . . 12 | |
18 | 17 | eceq1d 6465 | . . . . . . . . . . 11 |
19 | 18 | fveq2d 5425 | . . . . . . . . . 10 |
20 | 19 | oveq2d 5790 | . . . . . . . . 9 |
21 | 20 | breq2d 3941 | . . . . . . . 8 |
22 | 21 | abbidv 2257 | . . . . . . 7 |
23 | 20 | breq1d 3939 | . . . . . . . 8 |
24 | 23 | abbidv 2257 | . . . . . . 7 |
25 | 22, 24 | opeq12d 3713 | . . . . . 6 |
26 | fveq2 5421 | . . . . . 6 | |
27 | 25, 26 | breq12d 3942 | . . . . 5 |
28 | 27 | cbvrexv 2655 | . . . 4 |
29 | opeq1 3705 | . . . . . . . . . . . 12 | |
30 | 29 | eceq1d 6465 | . . . . . . . . . . 11 |
31 | 30 | fveq2d 5425 | . . . . . . . . . 10 |
32 | 31 | oveq2d 5790 | . . . . . . . . 9 |
33 | 32 | breq2d 3941 | . . . . . . . 8 |
34 | 33 | abbidv 2257 | . . . . . . 7 |
35 | 32 | breq1d 3939 | . . . . . . . 8 |
36 | 35 | abbidv 2257 | . . . . . . 7 |
37 | 34, 36 | opeq12d 3713 | . . . . . 6 |
38 | fveq2 5421 | . . . . . 6 | |
39 | 37, 38 | breq12d 3942 | . . . . 5 |
40 | 39 | cbvrexv 2655 | . . . 4 |
41 | 28, 40 | bitri 183 | . . 3 |
42 | 41 | anbi2i 452 | . 2 |
43 | 16, 42 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wcel 1480 cab 2125 wrex 2417 crab 2420 cop 3530 class class class wbr 3929 cfv 5123 (class class class)co 5774 c1st 6036 c1o 6306 cec 6427 cnpi 7080 ceq 7087 cnq 7088 cplq 7090 crq 7092 cltq 7093 cpp 7101 cltp 7103 |
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-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 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-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 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-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-1st 6038 df-ec 6431 df-qs 6435 df-ni 7112 df-nqqs 7156 |
This theorem is referenced by: caucvgprprlemopl 7505 caucvgprprlemlol 7506 caucvgprprlemdisj 7510 caucvgprprlemloc 7511 |
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