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Mirrors > Home > ILE Home > Th. List > dedekindicclemeu | Unicode version |
Description: Lemma for dedekindicc 13158. Part of proving uniqueness. (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 | |
dedekindicc.ab | |
dedekindicclemeu.are | |
dedekindicclemeu.ac | |
dedekindicclemeu.bre | |
dedekindicclemeu.bc | |
dedekindicclemeu.lt |
Ref | Expression |
---|---|
dedekindicclemeu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3979 | . . . 4 | |
2 | dedekindicclemeu.ac | . . . . . 6 | |
3 | 2 | simpld 111 | . . . . 5 |
4 | 3 | adantr 274 | . . . 4 |
5 | simpr 109 | . . . 4 | |
6 | 1, 4, 5 | rspcdva 2830 | . . 3 |
7 | dedekindicc.a | . . . . . . 7 | |
8 | dedekindicc.b | . . . . . . 7 | |
9 | iccssre 9882 | . . . . . . 7 | |
10 | 7, 8, 9 | syl2anc 409 | . . . . . 6 |
11 | dedekindicclemeu.are | . . . . . 6 | |
12 | 10, 11 | sseldd 3138 | . . . . 5 |
13 | 12 | ltnrd 8001 | . . . 4 |
14 | 13 | adantr 274 | . . 3 |
15 | 6, 14 | pm2.21fal 1362 | . 2 |
16 | breq2 3980 | . . . 4 | |
17 | dedekindicclemeu.bc | . . . . . 6 | |
18 | 17 | simprd 113 | . . . . 5 |
19 | 18 | adantr 274 | . . . 4 |
20 | simpr 109 | . . . 4 | |
21 | 16, 19, 20 | rspcdva 2830 | . . 3 |
22 | dedekindicclemeu.bre | . . . . . 6 | |
23 | 10, 22 | sseldd 3138 | . . . . 5 |
24 | 23 | ltnrd 8001 | . . . 4 |
25 | 24 | adantr 274 | . . 3 |
26 | 21, 25 | pm2.21fal 1362 | . 2 |
27 | dedekindicclemeu.lt | . . 3 | |
28 | breq2 3980 | . . . . 5 | |
29 | eleq1 2227 | . . . . . 6 | |
30 | 29 | orbi2d 780 | . . . . 5 |
31 | 28, 30 | imbi12d 233 | . . . 4 |
32 | breq1 3979 | . . . . . . 7 | |
33 | eleq1 2227 | . . . . . . . 8 | |
34 | 33 | orbi1d 781 | . . . . . . 7 |
35 | 32, 34 | imbi12d 233 | . . . . . 6 |
36 | 35 | ralbidv 2464 | . . . . 5 |
37 | dedekindicc.loc | . . . . 5 | |
38 | 36, 37, 11 | rspcdva 2830 | . . . 4 |
39 | 31, 38, 22 | rspcdva 2830 | . . 3 |
40 | 27, 39 | mpd 13 | . 2 |
41 | 15, 26, 40 | mpjaodan 788 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 wceq 1342 wfal 1347 wcel 2135 wral 2442 wrex 2443 cin 3110 wss 3111 c0 3404 class class class wbr 3976 (class class class)co 5836 cr 7743 clt 7924 cicc 9818 |
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 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 |
This theorem depends on definitions: df-bi 116 df-3or 968 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 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-icc 9822 |
This theorem is referenced by: dedekindicclemicc 13157 |
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