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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 4527 | . 2 | |
2 | noel 3362 | . . . . 5 | |
3 | eleq2 2201 | . . . . 5 | |
4 | 2, 3 | mtbiri 664 | . . . 4 |
5 | nner 2310 | . . . 4 | |
6 | 4, 5 | 2falsed 691 | . . 3 |
7 | id 19 | . . . 4 | |
8 | ne0i 3364 | . . . 4 | |
9 | 7, 8 | 2thd 174 | . . 3 |
10 | 6, 9 | jaoi 705 | . 2 |
11 | 1, 10 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wo 697 wceq 1331 wcel 1480 wne 2306 c0 3358 com 4499 |
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 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-suc 4288 df-iom 4500 |
This theorem is referenced by: nnmord 6406 nnnninf 7016 |
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