Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nn0eln0 | GIF 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 | ⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elnn 4532 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | |
2 | noel 3367 | . . . . 5 ⊢ ¬ ∅ ∈ ∅ | |
3 | eleq2 2203 | . . . . 5 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ ∅ ∈ ∅)) | |
4 | 2, 3 | mtbiri 664 | . . . 4 ⊢ (𝐴 = ∅ → ¬ ∅ ∈ 𝐴) |
5 | nner 2312 | . . . 4 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≠ ∅) | |
6 | 4, 5 | 2falsed 691 | . . 3 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
7 | id 19 | . . . 4 ⊢ (∅ ∈ 𝐴 → ∅ ∈ 𝐴) | |
8 | ne0i 3369 | . . . 4 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅) | |
9 | 7, 8 | 2thd 174 | . . 3 ⊢ (∅ ∈ 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
10 | 6, 9 | jaoi 705 | . 2 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
11 | 1, 10 | syl 14 | 1 ⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 ∅c0 3363 ωcom 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 ax-iinf 4502 |
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-ne 2309 df-ral 2421 df-rex 2422 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: nnmord 6413 nnnninf 7023 |
Copyright terms: Public domain | W3C validator |