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Mirrors > Home > ILE Home > Th. List > dedekindicclemuub | Unicode version |
Description: Lemma for dedekindicc 13370. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.) |
Ref | Expression |
---|---|
dedekindicc.a | |
dedekindicc.b | |
dedekindicc.lss | |
dedekindicc.uss | |
dedekindicc.lm | |
dedekindicc.um | |
dedekindicc.lr | |
dedekindicc.ur | |
dedekindicc.disj | |
dedekindicc.loc | |
dedekindicclemuub.u |
Ref | Expression |
---|---|
dedekindicclemuub |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedekindicclemuub.u | . . 3 | |
2 | eleq1 2233 | . . . . 5 | |
3 | breq2 3991 | . . . . . 6 | |
4 | 3 | rexbidv 2471 | . . . . 5 |
5 | 2, 4 | bibi12d 234 | . . . 4 |
6 | dedekindicc.ur | . . . 4 | |
7 | dedekindicc.uss | . . . . 5 | |
8 | 7, 1 | sseldd 3148 | . . . 4 |
9 | 5, 6, 8 | rspcdva 2839 | . . 3 |
10 | 1, 9 | mpbid 146 | . 2 |
11 | dedekindicc.a | . . . . . . 7 | |
12 | dedekindicc.b | . . . . . . 7 | |
13 | iccssre 9905 | . . . . . . 7 | |
14 | 11, 12, 13 | syl2anc 409 | . . . . . 6 |
15 | 14 | ad2antrr 485 | . . . . 5 |
16 | dedekindicc.lss | . . . . . . 7 | |
17 | 16 | ad2antrr 485 | . . . . . 6 |
18 | simpr 109 | . . . . . 6 | |
19 | 17, 18 | sseldd 3148 | . . . . 5 |
20 | 15, 19 | sseldd 3148 | . . . 4 |
21 | 7 | ad2antrr 485 | . . . . . 6 |
22 | simplrl 530 | . . . . . 6 | |
23 | 21, 22 | sseldd 3148 | . . . . 5 |
24 | 15, 23 | sseldd 3148 | . . . 4 |
25 | 14, 8 | sseldd 3148 | . . . . 5 |
26 | 25 | ad2antrr 485 | . . . 4 |
27 | breq1 3990 | . . . . . . . . . 10 | |
28 | 27 | rspcev 2834 | . . . . . . . . 9 |
29 | 22, 28 | sylan 281 | . . . . . . . 8 |
30 | 27 | cbvrexv 2697 | . . . . . . . 8 |
31 | 29, 30 | sylib 121 | . . . . . . 7 |
32 | eleq1 2233 | . . . . . . . . 9 | |
33 | breq2 3991 | . . . . . . . . . 10 | |
34 | 33 | rexbidv 2471 | . . . . . . . . 9 |
35 | 32, 34 | bibi12d 234 | . . . . . . . 8 |
36 | 6 | ad3antrrr 489 | . . . . . . . 8 |
37 | 19 | adantr 274 | . . . . . . . 8 |
38 | 35, 36, 37 | rspcdva 2839 | . . . . . . 7 |
39 | 31, 38 | mpbird 166 | . . . . . 6 |
40 | simplll 528 | . . . . . . 7 | |
41 | 18 | adantr 274 | . . . . . . 7 |
42 | dedekindicc.disj | . . . . . . . . 9 | |
43 | disj 3462 | . . . . . . . . 9 | |
44 | 42, 43 | sylib 121 | . . . . . . . 8 |
45 | 44 | r19.21bi 2558 | . . . . . . 7 |
46 | 40, 41, 45 | syl2anc 409 | . . . . . 6 |
47 | 39, 46 | pm2.65da 656 | . . . . 5 |
48 | 20, 24, 47 | nltled 8033 | . . . 4 |
49 | simplrr 531 | . . . 4 | |
50 | 20, 24, 26, 48, 49 | lelttrd 8037 | . . 3 |
51 | 50 | ralrimiva 2543 | . 2 |
52 | 10, 51 | rexlimddv 2592 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 wceq 1348 wcel 2141 wral 2448 wrex 2449 cin 3120 wss 3121 c0 3414 class class class wbr 3987 (class class class)co 5851 cr 7766 clt 7947 cicc 9841 |
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-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 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-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-icc 9845 |
This theorem is referenced by: dedekindicclemub 13364 dedekindicclemloc 13365 |
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