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Mirrors > Home > ILE Home > Th. List > nnregexmid | Unicode version |
Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4450 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6395 or nntri3or 6389), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.) |
Ref | Expression |
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nnregexmid.1 |
Ref | Expression |
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nnregexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3182 | . . . 4 | |
2 | peano1 4508 | . . . . 5 | |
3 | suc0 4333 | . . . . . 6 | |
4 | peano2 4509 | . . . . . . 7 | |
5 | 2, 4 | ax-mp 5 | . . . . . 6 |
6 | 3, 5 | eqeltrri 2213 | . . . . 5 |
7 | prssi 3678 | . . . . 5 | |
8 | 2, 6, 7 | mp2an 422 | . . . 4 |
9 | 1, 8 | sstri 3106 | . . 3 |
10 | eqid 2139 | . . . 4 | |
11 | 10 | regexmidlemm 4447 | . . 3 |
12 | pp0ex 4113 | . . . . 5 | |
13 | 12 | rabex 4072 | . . . 4 |
14 | sseq1 3120 | . . . . . 6 | |
15 | eleq2 2203 | . . . . . . 7 | |
16 | 15 | exbidv 1797 | . . . . . 6 |
17 | 14, 16 | anbi12d 464 | . . . . 5 |
18 | eleq2 2203 | . . . . . . . . . 10 | |
19 | 18 | notbid 656 | . . . . . . . . 9 |
20 | 19 | imbi2d 229 | . . . . . . . 8 |
21 | 20 | albidv 1796 | . . . . . . 7 |
22 | 15, 21 | anbi12d 464 | . . . . . 6 |
23 | 22 | exbidv 1797 | . . . . 5 |
24 | 17, 23 | imbi12d 233 | . . . 4 |
25 | nnregexmid.1 | . . . 4 | |
26 | 13, 24, 25 | vtocl 2740 | . . 3 |
27 | 9, 11, 26 | mp2an 422 | . 2 |
28 | 10 | regexmidlem1 4448 | . 2 |
29 | 27, 28 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 wal 1329 wceq 1331 wex 1468 wcel 1480 crab 2420 wss 3071 c0 3363 csn 3527 cpr 3528 csuc 4287 com 4504 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 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-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-suc 4293 df-iom 4505 |
This theorem is referenced by: (None) |
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