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Mirrors > Home > ILE Home > Th. List > dedekindeulemuub | Unicode version |
Description: Lemma for dedekindeu 12770. 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 2202 | . . . . 5 | |
3 | breq2 3933 | . . . . . 6 | |
4 | 3 | rexbidv 2438 | . . . . 5 |
5 | 2, 4 | bibi12d 234 | . . . 4 |
6 | dedekindeu.ur | . . . 4 | |
7 | dedekindeu.uss | . . . . 5 | |
8 | 7, 1 | sseldd 3098 | . . . 4 |
9 | 5, 6, 8 | rspcdva 2794 | . . 3 |
10 | 1, 9 | mpbid 146 | . 2 |
11 | dedekindeu.lss | . . . . . 6 | |
12 | 11 | ad2antrr 479 | . . . . 5 |
13 | simpr 109 | . . . . 5 | |
14 | 12, 13 | sseldd 3098 | . . . 4 |
15 | 7 | ad2antrr 479 | . . . . 5 |
16 | simplrl 524 | . . . . 5 | |
17 | 15, 16 | sseldd 3098 | . . . 4 |
18 | 8 | ad2antrr 479 | . . . 4 |
19 | breq1 3932 | . . . . . . . . . 10 | |
20 | 19 | rspcev 2789 | . . . . . . . . 9 |
21 | 16, 20 | sylan 281 | . . . . . . . 8 |
22 | 19 | cbvrexv 2655 | . . . . . . . 8 |
23 | 21, 22 | sylib 121 | . . . . . . 7 |
24 | eleq1 2202 | . . . . . . . . 9 | |
25 | breq2 3933 | . . . . . . . . . 10 | |
26 | 25 | rexbidv 2438 | . . . . . . . . 9 |
27 | 24, 26 | bibi12d 234 | . . . . . . . 8 |
28 | 6 | ad3antrrr 483 | . . . . . . . 8 |
29 | 14 | adantr 274 | . . . . . . . 8 |
30 | 27, 28, 29 | rspcdva 2794 | . . . . . . 7 |
31 | 23, 30 | mpbird 166 | . . . . . 6 |
32 | simplll 522 | . . . . . . 7 | |
33 | 13 | adantr 274 | . . . . . . 7 |
34 | dedekindeu.disj | . . . . . . . . 9 | |
35 | disj 3411 | . . . . . . . . 9 | |
36 | 34, 35 | sylib 121 | . . . . . . . 8 |
37 | 36 | r19.21bi 2520 | . . . . . . 7 |
38 | 32, 33, 37 | syl2anc 408 | . . . . . 6 |
39 | 31, 38 | pm2.65da 650 | . . . . 5 |
40 | 14, 17, 39 | nltled 7883 | . . . 4 |
41 | simplrr 525 | . . . 4 | |
42 | 14, 17, 18, 40, 41 | lelttrd 7887 | . . 3 |
43 | 42 | ralrimiva 2505 | . 2 |
44 | 10, 43 | rexlimddv 2554 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 wral 2416 wrex 2417 cin 3070 wss 3071 c0 3363 class class class wbr 3929 cr 7619 clt 7800 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltwlin 7733 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 |
This theorem is referenced by: dedekindeulemub 12765 dedekindeulemloc 12766 |
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