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Theorem nn0eln0 4341
 Description: A natural number is nonempty iff it contains the empty set. Although in constructive mathematics it is generally more natural to work with inhabited sets and ignore the whole concept of nonempty sets, in the specific case of natural numbers this theorem may be helpful in converting proofs which were written assuming excluded middle. (Contributed by Jim Kingdon, 28-Aug-2019.)
Assertion
Ref Expression
nn0eln0

Proof of Theorem nn0eln0
StepHypRef Expression
1 0elnn 4340 . 2
2 noel 3228 . . . . 5
3 eleq2 2101 . . . . 5
42, 3mtbiri 600 . . . 4
5 nner 2210 . . . 4
64, 52falsed 618 . . 3
7 id 19 . . . 4
8 ne0i 3230 . . . 4
97, 82thd 164 . . 3
106, 9jaoi 636 . 2
111, 10syl 14 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 98   wo 629   wceq 1243   wcel 1393   wne 2204  c0 3224  com 4313 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-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314 This theorem is referenced by:  nnmord  6090
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