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Mirrors > Home > ILE Home > Th. List > reg2exmidlema | Unicode version |
Description: Lemma for reg2exmid 4451. 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 519 | . . . . . . 7 | |
2 | sseq1 3120 | . . . . . . . . 9 | |
3 | 2 | ralbidv 2437 | . . . . . . . 8 |
4 | 3 | adantl 275 | . . . . . . 7 |
5 | 1, 4 | mpbid 146 | . . . . . 6 |
6 | 0ex 4055 | . . . . . . . 8 | |
7 | 6 | snss 3649 | . . . . . . 7 |
8 | 7 | ralbii 2441 | . . . . . 6 |
9 | 5, 8 | sylibr 133 | . . . . 5 |
10 | noel 3367 | . . . . . 6 | |
11 | eqid 2139 | . . . . . . . . . . . 12 | |
12 | 11 | jctl 312 | . . . . . . . . . . 11 |
13 | 12 | olcd 723 | . . . . . . . . . 10 |
14 | 6 | prid1 3629 | . . . . . . . . . 10 |
15 | 13, 14 | jctil 310 | . . . . . . . . 9 |
16 | eqeq1 2146 | . . . . . . . . . . 11 | |
17 | eqeq1 2146 | . . . . . . . . . . . 12 | |
18 | 17 | anbi1d 460 | . . . . . . . . . . 11 |
19 | 16, 18 | orbi12d 782 | . . . . . . . . . 10 |
20 | regexmidlemm.a | . . . . . . . . . 10 | |
21 | 19, 20 | elrab2 2843 | . . . . . . . . 9 |
22 | 15, 21 | sylibr 133 | . . . . . . . 8 |
23 | eleq2 2203 | . . . . . . . . 9 | |
24 | 23 | rspcv 2785 | . . . . . . . 8 |
25 | 22, 24 | syl 14 | . . . . . . 7 |
26 | 25 | com12 30 | . . . . . 6 |
27 | 10, 26 | mtoi 653 | . . . . 5 |
28 | 9, 27 | syl 14 | . . . 4 |
29 | 28 | olcd 723 | . . 3 |
30 | simprr 521 | . . . 4 | |
31 | 30 | orcd 722 | . . 3 |
32 | eqeq1 2146 | . . . . . . 7 | |
33 | eqeq1 2146 | . . . . . . . 8 | |
34 | 33 | anbi1d 460 | . . . . . . 7 |
35 | 32, 34 | orbi12d 782 | . . . . . 6 |
36 | 35, 20 | elrab2 2843 | . . . . 5 |
37 | 36 | simprbi 273 | . . . 4 |
38 | 37 | adantr 274 | . . 3 |
39 | 29, 31, 38 | mpjaodan 787 | . 2 |
40 | 39 | rexlimiva 2544 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 wral 2416 wrex 2417 crab 2420 wss 3071 c0 3363 csn 3527 cpr 3528 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-nul 4054 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 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-nul 3364 df-sn 3533 df-pr 3534 |
This theorem is referenced by: reg2exmid 4451 reg3exmid 4494 |
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