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Mirrors > Home > ILE Home > Th. List > caucvgprprlemell | Unicode version |
Description: Lemma for caucvgprpr 7544. 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 5789 | . . . . . . . 8 | |
2 | 1 | breq2d 3949 | . . . . . . 7 |
3 | 2 | abbidv 2258 | . . . . . 6 |
4 | 1 | breq1d 3947 | . . . . . . 7 |
5 | 4 | abbidv 2258 | . . . . . 6 |
6 | 3, 5 | opeq12d 3721 | . . . . 5 |
7 | 6 | breq1d 3947 | . . . 4 |
8 | 7 | rexbidv 2439 | . . 3 |
9 | caucvgprprlemell.lim | . . . . 5 | |
10 | 9 | fveq2i 5432 | . . . 4 |
11 | nqex 7195 | . . . . . 6 | |
12 | 11 | rabex 4080 | . . . . 5 |
13 | 11 | rabex 4080 | . . . . 5 |
14 | 12, 13 | op1st 6052 | . . . 4 |
15 | 10, 14 | eqtri 2161 | . . 3 |
16 | 8, 15 | elrab2 2847 | . 2 |
17 | opeq1 3713 | . . . . . . . . . . . 12 | |
18 | 17 | eceq1d 6473 | . . . . . . . . . . 11 |
19 | 18 | fveq2d 5433 | . . . . . . . . . 10 |
20 | 19 | oveq2d 5798 | . . . . . . . . 9 |
21 | 20 | breq2d 3949 | . . . . . . . 8 |
22 | 21 | abbidv 2258 | . . . . . . 7 |
23 | 20 | breq1d 3947 | . . . . . . . 8 |
24 | 23 | abbidv 2258 | . . . . . . 7 |
25 | 22, 24 | opeq12d 3721 | . . . . . 6 |
26 | fveq2 5429 | . . . . . 6 | |
27 | 25, 26 | breq12d 3950 | . . . . 5 |
28 | 27 | cbvrexv 2658 | . . . 4 |
29 | opeq1 3713 | . . . . . . . . . . . 12 | |
30 | 29 | eceq1d 6473 | . . . . . . . . . . 11 |
31 | 30 | fveq2d 5433 | . . . . . . . . . 10 |
32 | 31 | oveq2d 5798 | . . . . . . . . 9 |
33 | 32 | breq2d 3949 | . . . . . . . 8 |
34 | 33 | abbidv 2258 | . . . . . . 7 |
35 | 32 | breq1d 3947 | . . . . . . . 8 |
36 | 35 | abbidv 2258 | . . . . . . 7 |
37 | 34, 36 | opeq12d 3721 | . . . . . 6 |
38 | fveq2 5429 | . . . . . 6 | |
39 | 37, 38 | breq12d 3950 | . . . . 5 |
40 | 39 | cbvrexv 2658 | . . . 4 |
41 | 28, 40 | bitri 183 | . . 3 |
42 | 41 | anbi2i 453 | . 2 |
43 | 16, 42 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1332 wcel 1481 cab 2126 wrex 2418 crab 2421 cop 3535 class class class wbr 3937 cfv 5131 (class class class)co 5782 c1st 6044 c1o 6314 cec 6435 cnpi 7104 ceq 7111 cnq 7112 cplq 7114 crq 7116 cltq 7117 cpp 7125 cltp 7127 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-1st 6046 df-ec 6439 df-qs 6443 df-ni 7136 df-nqqs 7180 |
This theorem is referenced by: caucvgprprlemopl 7529 caucvgprprlemlol 7530 caucvgprprlemdisj 7534 caucvgprprlemloc 7535 |
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