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Mirrors > Home > ILE Home > Th. List > regexmidlem1 | Unicode version |
Description: Lemma for regexmid 4517. 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 2177 | . . . . . . 7 | |
2 | eqeq1 2177 | . . . . . . . 8 | |
3 | 2 | anbi1d 462 | . . . . . . 7 |
4 | 1, 3 | orbi12d 788 | . . . . . 6 |
5 | regexmidlemm.a | . . . . . 6 | |
6 | 4, 5 | elrab2 2889 | . . . . 5 |
7 | 6 | simprbi 273 | . . . 4 |
8 | 0ex 4114 | . . . . . . . . 9 | |
9 | 8 | snid 3612 | . . . . . . . 8 |
10 | eleq2 2234 | . . . . . . . 8 | |
11 | 9, 10 | mpbiri 167 | . . . . . . 7 |
12 | eleq1 2233 | . . . . . . . . 9 | |
13 | eleq1 2233 | . . . . . . . . . 10 | |
14 | 13 | notbid 662 | . . . . . . . . 9 |
15 | 12, 14 | imbi12d 233 | . . . . . . . 8 |
16 | 8, 15 | spcv 2824 | . . . . . . 7 |
17 | 11, 16 | syl5com 29 | . . . . . 6 |
18 | 8 | prid1 3687 | . . . . . . . . . 10 |
19 | eqeq1 2177 | . . . . . . . . . . . 12 | |
20 | eqeq1 2177 | . . . . . . . . . . . . 13 | |
21 | 20 | anbi1d 462 | . . . . . . . . . . . 12 |
22 | 19, 21 | orbi12d 788 | . . . . . . . . . . 11 |
23 | 22, 5 | elrab2 2889 | . . . . . . . . . 10 |
24 | 18, 23 | mpbiran 935 | . . . . . . . . 9 |
25 | pm2.46 734 | . . . . . . . . 9 | |
26 | 24, 25 | sylnbi 673 | . . . . . . . 8 |
27 | eqid 2170 | . . . . . . . . 9 | |
28 | 27 | biantrur 301 | . . . . . . . 8 |
29 | 26, 28 | sylnibr 672 | . . . . . . 7 |
30 | 29 | olcd 729 | . . . . . 6 |
31 | 17, 30 | syl6 33 | . . . . 5 |
32 | orc 707 | . . . . . . 7 | |
33 | 32 | adantl 275 | . . . . . 6 |
34 | 33 | a1d 22 | . . . . 5 |
35 | 31, 34 | jaoi 711 | . . . 4 |
36 | 7, 35 | syl 14 | . . 3 |
37 | 36 | imp 123 | . 2 |
38 | 37 | exlimiv 1591 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 wal 1346 wceq 1348 wex 1485 wcel 2141 crab 2452 c0 3414 csn 3581 cpr 3582 |
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-ext 2152 ax-nul 4113 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-nul 3415 df-sn 3587 df-pr 3588 |
This theorem is referenced by: regexmid 4517 nnregexmid 4603 |
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