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Theorem nnregexmid 4534
 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.)
Hypothesis
Ref Expression
nnregexmid.1
Assertion
Ref Expression
nnregexmid
Distinct variable group:   ,,,

Proof of Theorem nnregexmid
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3182 . . . 4
2 peano1 4508 . . . . 5
3 suc0 4333 . . . . . 6
4 peano2 4509 . . . . . . 7
52, 4ax-mp 5 . . . . . 6
63, 5eqeltrri 2213 . . . . 5
7 prssi 3678 . . . . 5
82, 6, 7mp2an 422 . . . 4
91, 8sstri 3106 . . 3
10 eqid 2139 . . . 4
1110regexmidlemm 4447 . . 3
12 pp0ex 4113 . . . . 5
1312rabex 4072 . . . 4
14 sseq1 3120 . . . . . 6
15 eleq2 2203 . . . . . . 7
1615exbidv 1797 . . . . . 6
1714, 16anbi12d 464 . . . . 5
18 eleq2 2203 . . . . . . . . . 10
1918notbid 656 . . . . . . . . 9
2019imbi2d 229 . . . . . . . 8
2120albidv 1796 . . . . . . 7
2215, 21anbi12d 464 . . . . . 6
2322exbidv 1797 . . . . 5
2417, 23imbi12d 233 . . . 4
25 nnregexmid.1 . . . 4
2613, 24, 25vtocl 2740 . . 3
279, 11, 26mp2an 422 . 2
2810regexmidlem1 4448 . 2
2927, 28ax-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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