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Mirrors > Home > ILE Home > Th. List > regexmidlem1 | Unicode version |
Description: Lemma for regexmid 4260. If has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Ref | Expression |
---|---|
regexmidlemm.a |
Ref | Expression |
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regexmidlem1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2046 | . . . . . . 7 | |
2 | eqeq1 2046 | . . . . . . . 8 | |
3 | 2 | anbi1d 438 | . . . . . . 7 |
4 | 1, 3 | orbi12d 707 | . . . . . 6 |
5 | regexmidlemm.a | . . . . . 6 | |
6 | 4, 5 | elrab2 2700 | . . . . 5 |
7 | 6 | simprbi 260 | . . . 4 |
8 | 0ex 3884 | . . . . . . . . 9 | |
9 | 8 | snid 3402 | . . . . . . . 8 |
10 | eleq2 2101 | . . . . . . . 8 | |
11 | 9, 10 | mpbiri 157 | . . . . . . 7 |
12 | eleq1 2100 | . . . . . . . . 9 | |
13 | eleq1 2100 | . . . . . . . . . 10 | |
14 | 13 | notbid 592 | . . . . . . . . 9 |
15 | 12, 14 | imbi12d 223 | . . . . . . . 8 |
16 | 8, 15 | spcv 2646 | . . . . . . 7 |
17 | 11, 16 | syl5com 26 | . . . . . 6 |
18 | 8 | prid1 3476 | . . . . . . . . . 10 |
19 | eqeq1 2046 | . . . . . . . . . . . 12 | |
20 | eqeq1 2046 | . . . . . . . . . . . . 13 | |
21 | 20 | anbi1d 438 | . . . . . . . . . . . 12 |
22 | 19, 21 | orbi12d 707 | . . . . . . . . . . 11 |
23 | 22, 5 | elrab2 2700 | . . . . . . . . . 10 |
24 | 18, 23 | mpbiran 847 | . . . . . . . . 9 |
25 | pm2.46 658 | . . . . . . . . 9 | |
26 | 24, 25 | sylnbi 603 | . . . . . . . 8 |
27 | eqid 2040 | . . . . . . . . 9 | |
28 | 27 | biantrur 287 | . . . . . . . 8 |
29 | 26, 28 | sylnibr 602 | . . . . . . 7 |
30 | 29 | olcd 653 | . . . . . 6 |
31 | 17, 30 | syl6 29 | . . . . 5 |
32 | orc 633 | . . . . . . 7 | |
33 | 32 | adantl 262 | . . . . . 6 |
34 | 33 | a1d 22 | . . . . 5 |
35 | 31, 34 | jaoi 636 | . . . 4 |
36 | 7, 35 | syl 14 | . . 3 |
37 | 36 | imp 115 | . 2 |
38 | 37 | exlimiv 1489 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wo 629 wal 1241 wceq 1243 wex 1381 wcel 1393 crab 2310 c0 3224 csn 3375 cpr 3376 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-nul 3883 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-nul 3225 df-sn 3381 df-pr 3382 |
This theorem is referenced by: regexmid 4260 nnregexmid 4342 |
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