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Mirrors > Home > ILE Home > Th. List > dedekindeulemuub | Unicode version |
Description: Lemma for dedekindeu 13168. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.) |
Ref | Expression |
---|---|
dedekindeu.lss | |
dedekindeu.uss | |
dedekindeu.lm | |
dedekindeu.um | |
dedekindeu.lr | |
dedekindeu.ur | |
dedekindeu.disj | |
dedekindeu.loc | |
dedekindeulemuub.u |
Ref | Expression |
---|---|
dedekindeulemuub |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedekindeulemuub.u | . . 3 | |
2 | eleq1 2227 | . . . . 5 | |
3 | breq2 3981 | . . . . . 6 | |
4 | 3 | rexbidv 2465 | . . . . 5 |
5 | 2, 4 | bibi12d 234 | . . . 4 |
6 | dedekindeu.ur | . . . 4 | |
7 | dedekindeu.uss | . . . . 5 | |
8 | 7, 1 | sseldd 3139 | . . . 4 |
9 | 5, 6, 8 | rspcdva 2831 | . . 3 |
10 | 1, 9 | mpbid 146 | . 2 |
11 | dedekindeu.lss | . . . . . 6 | |
12 | 11 | ad2antrr 480 | . . . . 5 |
13 | simpr 109 | . . . . 5 | |
14 | 12, 13 | sseldd 3139 | . . . 4 |
15 | 7 | ad2antrr 480 | . . . . 5 |
16 | simplrl 525 | . . . . 5 | |
17 | 15, 16 | sseldd 3139 | . . . 4 |
18 | 8 | ad2antrr 480 | . . . 4 |
19 | breq1 3980 | . . . . . . . . . 10 | |
20 | 19 | rspcev 2826 | . . . . . . . . 9 |
21 | 16, 20 | sylan 281 | . . . . . . . 8 |
22 | 19 | cbvrexv 2691 | . . . . . . . 8 |
23 | 21, 22 | sylib 121 | . . . . . . 7 |
24 | eleq1 2227 | . . . . . . . . 9 | |
25 | breq2 3981 | . . . . . . . . . 10 | |
26 | 25 | rexbidv 2465 | . . . . . . . . 9 |
27 | 24, 26 | bibi12d 234 | . . . . . . . 8 |
28 | 6 | ad3antrrr 484 | . . . . . . . 8 |
29 | 14 | adantr 274 | . . . . . . . 8 |
30 | 27, 28, 29 | rspcdva 2831 | . . . . . . 7 |
31 | 23, 30 | mpbird 166 | . . . . . 6 |
32 | simplll 523 | . . . . . . 7 | |
33 | 13 | adantr 274 | . . . . . . 7 |
34 | dedekindeu.disj | . . . . . . . . 9 | |
35 | disj 3453 | . . . . . . . . 9 | |
36 | 34, 35 | sylib 121 | . . . . . . . 8 |
37 | 36 | r19.21bi 2552 | . . . . . . 7 |
38 | 32, 33, 37 | syl2anc 409 | . . . . . 6 |
39 | 31, 38 | pm2.65da 651 | . . . . 5 |
40 | 14, 17, 39 | nltled 8011 | . . . 4 |
41 | simplrr 526 | . . . 4 | |
42 | 14, 17, 18, 40, 41 | lelttrd 8015 | . . 3 |
43 | 42 | ralrimiva 2537 | . 2 |
44 | 10, 43 | rexlimddv 2586 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 wceq 1342 wcel 2135 wral 2442 wrex 2443 cin 3111 wss 3112 c0 3405 class class class wbr 3977 cr 7744 clt 7925 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-cnex 7836 ax-resscn 7837 ax-pre-ltwlin 7858 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2724 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-opab 4039 df-xp 4605 df-cnv 4607 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 |
This theorem is referenced by: dedekindeulemub 13163 dedekindeulemloc 13164 |
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