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Mirrors > Home > ILE Home > Th. List > reg2exmidlema | Unicode version |
Description: Lemma for reg2exmid 4520. 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 525 | . . . . . . 7 | |
2 | sseq1 3170 | . . . . . . . . 9 | |
3 | 2 | ralbidv 2470 | . . . . . . . 8 |
4 | 3 | adantl 275 | . . . . . . 7 |
5 | 1, 4 | mpbid 146 | . . . . . 6 |
6 | 0ex 4116 | . . . . . . . 8 | |
7 | 6 | snss 3709 | . . . . . . 7 |
8 | 7 | ralbii 2476 | . . . . . 6 |
9 | 5, 8 | sylibr 133 | . . . . 5 |
10 | noel 3418 | . . . . . 6 | |
11 | eqid 2170 | . . . . . . . . . . . 12 | |
12 | 11 | jctl 312 | . . . . . . . . . . 11 |
13 | 12 | olcd 729 | . . . . . . . . . 10 |
14 | 6 | prid1 3689 | . . . . . . . . . 10 |
15 | 13, 14 | jctil 310 | . . . . . . . . 9 |
16 | eqeq1 2177 | . . . . . . . . . . 11 | |
17 | eqeq1 2177 | . . . . . . . . . . . 12 | |
18 | 17 | anbi1d 462 | . . . . . . . . . . 11 |
19 | 16, 18 | orbi12d 788 | . . . . . . . . . 10 |
20 | regexmidlemm.a | . . . . . . . . . 10 | |
21 | 19, 20 | elrab2 2889 | . . . . . . . . 9 |
22 | 15, 21 | sylibr 133 | . . . . . . . 8 |
23 | eleq2 2234 | . . . . . . . . 9 | |
24 | 23 | rspcv 2830 | . . . . . . . 8 |
25 | 22, 24 | syl 14 | . . . . . . 7 |
26 | 25 | com12 30 | . . . . . 6 |
27 | 10, 26 | mtoi 659 | . . . . 5 |
28 | 9, 27 | syl 14 | . . . 4 |
29 | 28 | olcd 729 | . . 3 |
30 | simprr 527 | . . . 4 | |
31 | 30 | orcd 728 | . . 3 |
32 | eqeq1 2177 | . . . . . . 7 | |
33 | eqeq1 2177 | . . . . . . . 8 | |
34 | 33 | anbi1d 462 | . . . . . . 7 |
35 | 32, 34 | orbi12d 788 | . . . . . 6 |
36 | 35, 20 | elrab2 2889 | . . . . 5 |
37 | 36 | simprbi 273 | . . . 4 |
38 | 37 | adantr 274 | . . 3 |
39 | 29, 31, 38 | mpjaodan 793 | . 2 |
40 | 39 | rexlimiva 2582 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 wceq 1348 wcel 2141 wral 2448 wrex 2449 crab 2452 wss 3121 c0 3414 csn 3583 cpr 3584 |
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 4115 |
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-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-sn 3589 df-pr 3590 |
This theorem is referenced by: reg2exmid 4520 reg3exmid 4564 |
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