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Mirrors > Home > ILE Home > Th. List > caucvgprprlemell | Unicode version |
Description: Lemma for caucvgprpr 7674. 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 5860 | . . . . . . . 8 | |
2 | 1 | breq2d 4001 | . . . . . . 7 |
3 | 2 | abbidv 2288 | . . . . . 6 |
4 | 1 | breq1d 3999 | . . . . . . 7 |
5 | 4 | abbidv 2288 | . . . . . 6 |
6 | 3, 5 | opeq12d 3773 | . . . . 5 |
7 | 6 | breq1d 3999 | . . . 4 |
8 | 7 | rexbidv 2471 | . . 3 |
9 | caucvgprprlemell.lim | . . . . 5 | |
10 | 9 | fveq2i 5499 | . . . 4 |
11 | nqex 7325 | . . . . . 6 | |
12 | 11 | rabex 4133 | . . . . 5 |
13 | 11 | rabex 4133 | . . . . 5 |
14 | 12, 13 | op1st 6125 | . . . 4 |
15 | 10, 14 | eqtri 2191 | . . 3 |
16 | 8, 15 | elrab2 2889 | . 2 |
17 | opeq1 3765 | . . . . . . . . . . . 12 | |
18 | 17 | eceq1d 6549 | . . . . . . . . . . 11 |
19 | 18 | fveq2d 5500 | . . . . . . . . . 10 |
20 | 19 | oveq2d 5869 | . . . . . . . . 9 |
21 | 20 | breq2d 4001 | . . . . . . . 8 |
22 | 21 | abbidv 2288 | . . . . . . 7 |
23 | 20 | breq1d 3999 | . . . . . . . 8 |
24 | 23 | abbidv 2288 | . . . . . . 7 |
25 | 22, 24 | opeq12d 3773 | . . . . . 6 |
26 | fveq2 5496 | . . . . . 6 | |
27 | 25, 26 | breq12d 4002 | . . . . 5 |
28 | 27 | cbvrexv 2697 | . . . 4 |
29 | opeq1 3765 | . . . . . . . . . . . 12 | |
30 | 29 | eceq1d 6549 | . . . . . . . . . . 11 |
31 | 30 | fveq2d 5500 | . . . . . . . . . 10 |
32 | 31 | oveq2d 5869 | . . . . . . . . 9 |
33 | 32 | breq2d 4001 | . . . . . . . 8 |
34 | 33 | abbidv 2288 | . . . . . . 7 |
35 | 32 | breq1d 3999 | . . . . . . . 8 |
36 | 35 | abbidv 2288 | . . . . . . 7 |
37 | 34, 36 | opeq12d 3773 | . . . . . 6 |
38 | fveq2 5496 | . . . . . 6 | |
39 | 37, 38 | breq12d 4002 | . . . . 5 |
40 | 39 | cbvrexv 2697 | . . . 4 |
41 | 28, 40 | bitri 183 | . . 3 |
42 | 41 | anbi2i 454 | . 2 |
43 | 16, 42 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1348 wcel 2141 cab 2156 wrex 2449 crab 2452 cop 3586 class class class wbr 3989 cfv 5198 (class class class)co 5853 c1st 6117 c1o 6388 cec 6511 cnpi 7234 ceq 7241 cnq 7242 cplq 7244 crq 7246 cltq 7247 cpp 7255 cltp 7257 |
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-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 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-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 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-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-1st 6119 df-ec 6515 df-qs 6519 df-ni 7266 df-nqqs 7310 |
This theorem is referenced by: caucvgprprlemopl 7659 caucvgprprlemlol 7660 caucvgprprlemdisj 7664 caucvgprprlemloc 7665 |
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