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Mirrors > Home > ILE Home > Th. List > nn0eln0 | Unicode version |
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.) |
Ref | Expression |
---|---|
nn0eln0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elnn 4578 | . 2 | |
2 | noel 3398 | . . . . 5 | |
3 | eleq2 2221 | . . . . 5 | |
4 | 2, 3 | mtbiri 665 | . . . 4 |
5 | nner 2331 | . . . 4 | |
6 | 4, 5 | 2falsed 692 | . . 3 |
7 | id 19 | . . . 4 | |
8 | ne0i 3400 | . . . 4 | |
9 | 7, 8 | 2thd 174 | . . 3 |
10 | 6, 9 | jaoi 706 | . 2 |
11 | 1, 10 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wo 698 wceq 1335 wcel 2128 wne 2327 c0 3394 com 4549 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-uni 3773 df-int 3808 df-suc 4331 df-iom 4550 |
This theorem is referenced by: nnmord 6464 bj-charfunr 13396 |
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