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Mirrors > Home > ILE Home > Th. List > dedekindeulemub | Unicode version |
Description: Lemma for dedekindeu 12773. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.) |
Ref | Expression |
---|---|
dedekindeu.lss | |
dedekindeu.uss | |
dedekindeu.lm | |
dedekindeu.um | |
dedekindeu.lr | |
dedekindeu.ur | |
dedekindeu.disj | |
dedekindeu.loc |
Ref | Expression |
---|---|
dedekindeulemub |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedekindeu.um | . . 3 | |
2 | eleq1w 2200 | . . . 4 | |
3 | 2 | cbvrexv 2655 | . . 3 |
4 | 1, 3 | sylib 121 | . 2 |
5 | simprl 520 | . . 3 | |
6 | dedekindeu.lss | . . . . 5 | |
7 | 6 | adantr 274 | . . . 4 |
8 | dedekindeu.uss | . . . . 5 | |
9 | 8 | adantr 274 | . . . 4 |
10 | dedekindeu.lm | . . . . 5 | |
11 | 10 | adantr 274 | . . . 4 |
12 | 1 | adantr 274 | . . . 4 |
13 | dedekindeu.lr | . . . . 5 | |
14 | 13 | adantr 274 | . . . 4 |
15 | dedekindeu.ur | . . . . 5 | |
16 | 15 | adantr 274 | . . . 4 |
17 | dedekindeu.disj | . . . . 5 | |
18 | 17 | adantr 274 | . . . 4 |
19 | dedekindeu.loc | . . . . 5 | |
20 | 19 | adantr 274 | . . . 4 |
21 | simprr 521 | . . . 4 | |
22 | 7, 9, 11, 12, 14, 16, 18, 20, 21 | dedekindeulemuub 12767 | . . 3 |
23 | brralrspcev 3986 | . . 3 | |
24 | 5, 22, 23 | syl2anc 408 | . 2 |
25 | 4, 24 | rexlimddv 2554 | 1 |
Colors of variables: wff set class |
Syntax hints: 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 7622 clt 7803 |
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 7714 ax-resscn 7715 ax-pre-ltwlin 7736 |
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 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 |
This theorem is referenced by: dedekindeulemlub 12770 |
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