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Mirrors > Home > ILE Home > Th. List > reg2exmidlema | Unicode version |
Description: Lemma for reg2exmid 4261. If has a minimal element (expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Ref | Expression |
---|---|
regexmidlemm.a |
Ref | Expression |
---|---|
reg2exmidlema |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 482 | . . . . . . 7 | |
2 | sseq1 2966 | . . . . . . . . 9 | |
3 | 2 | ralbidv 2326 | . . . . . . . 8 |
4 | 3 | adantl 262 | . . . . . . 7 |
5 | 1, 4 | mpbid 135 | . . . . . 6 |
6 | 0ex 3884 | . . . . . . . 8 | |
7 | 6 | snss 3494 | . . . . . . 7 |
8 | 7 | ralbii 2330 | . . . . . 6 |
9 | 5, 8 | sylibr 137 | . . . . 5 |
10 | noel 3228 | . . . . . 6 | |
11 | eqid 2040 | . . . . . . . . . . . 12 | |
12 | 11 | jctl 297 | . . . . . . . . . . 11 |
13 | 12 | olcd 653 | . . . . . . . . . 10 |
14 | 6 | prid1 3476 | . . . . . . . . . 10 |
15 | 13, 14 | jctil 295 | . . . . . . . . 9 |
16 | eqeq1 2046 | . . . . . . . . . . 11 | |
17 | eqeq1 2046 | . . . . . . . . . . . 12 | |
18 | 17 | anbi1d 438 | . . . . . . . . . . 11 |
19 | 16, 18 | orbi12d 707 | . . . . . . . . . 10 |
20 | regexmidlemm.a | . . . . . . . . . 10 | |
21 | 19, 20 | elrab2 2700 | . . . . . . . . 9 |
22 | 15, 21 | sylibr 137 | . . . . . . . 8 |
23 | eleq2 2101 | . . . . . . . . 9 | |
24 | 23 | rspcv 2652 | . . . . . . . 8 |
25 | 22, 24 | syl 14 | . . . . . . 7 |
26 | 25 | com12 27 | . . . . . 6 |
27 | 10, 26 | mtoi 590 | . . . . 5 |
28 | 9, 27 | syl 14 | . . . 4 |
29 | 28 | olcd 653 | . . 3 |
30 | simprr 484 | . . . 4 | |
31 | 30 | orcd 652 | . . 3 |
32 | eqeq1 2046 | . . . . . . 7 | |
33 | eqeq1 2046 | . . . . . . . 8 | |
34 | 33 | anbi1d 438 | . . . . . . 7 |
35 | 32, 34 | orbi12d 707 | . . . . . 6 |
36 | 35, 20 | elrab2 2700 | . . . . 5 |
37 | 36 | simprbi 260 | . . . 4 |
38 | 37 | adantr 261 | . . 3 |
39 | 29, 31, 38 | mpjaodan 711 | . 2 |
40 | 39 | rexlimiva 2428 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 wceq 1243 wcel 1393 wral 2306 wrex 2307 crab 2310 wss 2917 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-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-sn 3381 df-pr 3382 |
This theorem is referenced by: reg2exmid 4261 reg3exmid 4304 |
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