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Mirrors > Home > ILE Home > Th. List > dedekindeulemloc | Unicode version |
Description: Lemma for dedekindeu 13395. The set L is located. (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 |
---|---|
dedekindeulemloc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 3993 | . . . . 5 | |
2 | eleq1w 2231 | . . . . . 6 | |
3 | 2 | orbi2d 785 | . . . . 5 |
4 | 1, 3 | imbi12d 233 | . . . 4 |
5 | breq1 3992 | . . . . . . 7 | |
6 | eleq1w 2231 | . . . . . . . 8 | |
7 | 6 | orbi1d 786 | . . . . . . 7 |
8 | 5, 7 | imbi12d 233 | . . . . . 6 |
9 | 8 | ralbidv 2470 | . . . . 5 |
10 | dedekindeu.loc | . . . . . 6 | |
11 | 10 | adantr 274 | . . . . 5 |
12 | simprl 526 | . . . . 5 | |
13 | 9, 11, 12 | rspcdva 2839 | . . . 4 |
14 | simprr 527 | . . . 4 | |
15 | 4, 13, 14 | rspcdva 2839 | . . 3 |
16 | simpr 109 | . . . . . . 7 | |
17 | 5 | rexbidv 2471 | . . . . . . . . 9 |
18 | 6, 17 | bibi12d 234 | . . . . . . . 8 |
19 | dedekindeu.lr | . . . . . . . . 9 | |
20 | 19 | ad2antrr 485 | . . . . . . . 8 |
21 | 12 | adantr 274 | . . . . . . . 8 |
22 | 18, 20, 21 | rspcdva 2839 | . . . . . . 7 |
23 | 16, 22 | mpbid 146 | . . . . . 6 |
24 | breq2 3993 | . . . . . . 7 | |
25 | 24 | cbvrexv 2697 | . . . . . 6 |
26 | 23, 25 | sylib 121 | . . . . 5 |
27 | 26 | ex 114 | . . . 4 |
28 | dedekindeu.lss | . . . . . . 7 | |
29 | 28 | ad2antrr 485 | . . . . . 6 |
30 | dedekindeu.uss | . . . . . . 7 | |
31 | 30 | ad2antrr 485 | . . . . . 6 |
32 | dedekindeu.lm | . . . . . . 7 | |
33 | 32 | ad2antrr 485 | . . . . . 6 |
34 | dedekindeu.um | . . . . . . 7 | |
35 | 34 | ad2antrr 485 | . . . . . 6 |
36 | 19 | ad2antrr 485 | . . . . . 6 |
37 | dedekindeu.ur | . . . . . . 7 | |
38 | 37 | ad2antrr 485 | . . . . . 6 |
39 | dedekindeu.disj | . . . . . . 7 | |
40 | 39 | ad2antrr 485 | . . . . . 6 |
41 | 10 | ad2antrr 485 | . . . . . 6 |
42 | simpr 109 | . . . . . 6 | |
43 | 29, 31, 33, 35, 36, 38, 40, 41, 42 | dedekindeulemuub 13389 | . . . . 5 |
44 | 43 | ex 114 | . . . 4 |
45 | 27, 44 | orim12d 781 | . . 3 |
46 | 15, 45 | syld 45 | . 2 |
47 | 46 | ralrimivva 2552 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 703 wceq 1348 wcel 2141 wral 2448 wrex 2449 cin 3120 wss 3121 c0 3414 class class class wbr 3989 cr 7773 clt 7954 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltwlin 7887 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-cnv 4619 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 |
This theorem is referenced by: dedekindeulemlub 13392 |
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