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Mirrors > Home > ILE Home > Th. List > dedekindeulemeu | Unicode version |
Description: Lemma for dedekindeu 12770. Part of proving uniqueness. (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 | |
dedekindeulemeu.are | |
dedekindeulemeu.ac | |
dedekindeulemeu.bre | |
dedekindeulemeu.bc | |
dedekindeulemeu.lt |
Ref | Expression |
---|---|
dedekindeulemeu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3932 | . . . 4 | |
2 | dedekindeulemeu.ac | . . . . . 6 | |
3 | 2 | simpld 111 | . . . . 5 |
4 | 3 | adantr 274 | . . . 4 |
5 | simpr 109 | . . . 4 | |
6 | 1, 4, 5 | rspcdva 2794 | . . 3 |
7 | dedekindeulemeu.are | . . . . 5 | |
8 | 7 | ltnrd 7875 | . . . 4 |
9 | 8 | adantr 274 | . . 3 |
10 | 6, 9 | pm2.21fal 1351 | . 2 |
11 | breq2 3933 | . . . 4 | |
12 | dedekindeulemeu.bc | . . . . . 6 | |
13 | 12 | simprd 113 | . . . . 5 |
14 | 13 | adantr 274 | . . . 4 |
15 | simpr 109 | . . . 4 | |
16 | 11, 14, 15 | rspcdva 2794 | . . 3 |
17 | dedekindeulemeu.bre | . . . . 5 | |
18 | 17 | ltnrd 7875 | . . . 4 |
19 | 18 | adantr 274 | . . 3 |
20 | 16, 19 | pm2.21fal 1351 | . 2 |
21 | dedekindeulemeu.lt | . . 3 | |
22 | breq2 3933 | . . . . 5 | |
23 | eleq1 2202 | . . . . . 6 | |
24 | 23 | orbi2d 779 | . . . . 5 |
25 | 22, 24 | imbi12d 233 | . . . 4 |
26 | breq1 3932 | . . . . . . 7 | |
27 | eleq1 2202 | . . . . . . . 8 | |
28 | 27 | orbi1d 780 | . . . . . . 7 |
29 | 26, 28 | imbi12d 233 | . . . . . 6 |
30 | 29 | ralbidv 2437 | . . . . 5 |
31 | dedekindeu.loc | . . . . 5 | |
32 | 30, 31, 7 | rspcdva 2794 | . . . 4 |
33 | 25, 32, 17 | rspcdva 2794 | . . 3 |
34 | 21, 33 | mpd 13 | . 2 |
35 | 10, 20, 34 | mpjaodan 787 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wceq 1331 wfal 1336 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-ltirr 7732 |
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-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-ltxr 7805 |
This theorem is referenced by: dedekindeu 12770 |
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