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Mirrors > Home > ILE Home > Th. List > dedekindeulemub | Unicode version |
Description: Lemma for dedekindeu 13159. 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 2225 | . . . 4 | |
3 | 2 | cbvrexv 2691 | . . 3 |
4 | 1, 3 | sylib 121 | . 2 |
5 | simprl 521 | . . 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 522 | . . . 4 | |
22 | 7, 9, 11, 12, 14, 16, 18, 20, 21 | dedekindeulemuub 13153 | . . 3 |
23 | brralrspcev 4035 | . . 3 | |
24 | 5, 22, 23 | syl2anc 409 | . 2 |
25 | 4, 24 | rexlimddv 2586 | 1 |
Colors of variables: wff set class |
Syntax hints: 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: dedekindeulemlub 13156 |
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