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Mirrors > Home > ILE Home > Th. List > caucvgprprlemell | Unicode version |
Description: Lemma for caucvgprpr 7633. 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 5832 | . . . . . . . 8 | |
2 | 1 | breq2d 3978 | . . . . . . 7 |
3 | 2 | abbidv 2275 | . . . . . 6 |
4 | 1 | breq1d 3976 | . . . . . . 7 |
5 | 4 | abbidv 2275 | . . . . . 6 |
6 | 3, 5 | opeq12d 3750 | . . . . 5 |
7 | 6 | breq1d 3976 | . . . 4 |
8 | 7 | rexbidv 2458 | . . 3 |
9 | caucvgprprlemell.lim | . . . . 5 | |
10 | 9 | fveq2i 5472 | . . . 4 |
11 | nqex 7284 | . . . . . 6 | |
12 | 11 | rabex 4109 | . . . . 5 |
13 | 11 | rabex 4109 | . . . . 5 |
14 | 12, 13 | op1st 6095 | . . . 4 |
15 | 10, 14 | eqtri 2178 | . . 3 |
16 | 8, 15 | elrab2 2871 | . 2 |
17 | opeq1 3742 | . . . . . . . . . . . 12 | |
18 | 17 | eceq1d 6517 | . . . . . . . . . . 11 |
19 | 18 | fveq2d 5473 | . . . . . . . . . 10 |
20 | 19 | oveq2d 5841 | . . . . . . . . 9 |
21 | 20 | breq2d 3978 | . . . . . . . 8 |
22 | 21 | abbidv 2275 | . . . . . . 7 |
23 | 20 | breq1d 3976 | . . . . . . . 8 |
24 | 23 | abbidv 2275 | . . . . . . 7 |
25 | 22, 24 | opeq12d 3750 | . . . . . 6 |
26 | fveq2 5469 | . . . . . 6 | |
27 | 25, 26 | breq12d 3979 | . . . . 5 |
28 | 27 | cbvrexv 2681 | . . . 4 |
29 | opeq1 3742 | . . . . . . . . . . . 12 | |
30 | 29 | eceq1d 6517 | . . . . . . . . . . 11 |
31 | 30 | fveq2d 5473 | . . . . . . . . . 10 |
32 | 31 | oveq2d 5841 | . . . . . . . . 9 |
33 | 32 | breq2d 3978 | . . . . . . . 8 |
34 | 33 | abbidv 2275 | . . . . . . 7 |
35 | 32 | breq1d 3976 | . . . . . . . 8 |
36 | 35 | abbidv 2275 | . . . . . . 7 |
37 | 34, 36 | opeq12d 3750 | . . . . . 6 |
38 | fveq2 5469 | . . . . . 6 | |
39 | 37, 38 | breq12d 3979 | . . . . 5 |
40 | 39 | cbvrexv 2681 | . . . 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 1335 wcel 2128 cab 2143 wrex 2436 crab 2439 cop 3563 class class class wbr 3966 cfv 5171 (class class class)co 5825 c1st 6087 c1o 6357 cec 6479 cnpi 7193 ceq 7200 cnq 7201 cplq 7203 crq 7205 cltq 7206 cpp 7214 cltp 7216 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-ov 5828 df-1st 6089 df-ec 6483 df-qs 6487 df-ni 7225 df-nqqs 7269 |
This theorem is referenced by: caucvgprprlemopl 7618 caucvgprprlemlol 7619 caucvgprprlemdisj 7623 caucvgprprlemloc 7624 |
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