Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dedekindicclemuub | Unicode version |
Description: Lemma for dedekindicc 13405. 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 3993 | . . . . . 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 9912 | . . . . . . 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 3992 | . . . . . . . . . 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 3993 | . . . . . . . . . 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 3463 | . . . . . . . . 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 8040 | . . . 4 |
49 | simplrr 531 | . . . 4 | |
50 | 20, 24, 26, 48, 49 | lelttrd 8044 | . . 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 3989 (class class class)co 5853 cr 7773 clt 7954 cicc 9848 |
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 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 |
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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-icc 9852 |
This theorem is referenced by: dedekindicclemub 13399 dedekindicclemloc 13400 |
Copyright terms: Public domain | W3C validator |