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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 4633 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | |
2 | noel 3441 | . . . . 5 ⊢ ¬ ∅ ∈ ∅ | |
3 | eleq2 2253 | . . . . 5 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ ∅ ∈ ∅)) | |
4 | 2, 3 | mtbiri 676 | . . . 4 ⊢ (𝐴 = ∅ → ¬ ∅ ∈ 𝐴) |
5 | nner 2364 | . . . 4 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≠ ∅) | |
6 | 4, 5 | 2falsed 703 | . . 3 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
7 | id 19 | . . . 4 ⊢ (∅ ∈ 𝐴 → ∅ ∈ 𝐴) | |
8 | ne0i 3444 | . . . 4 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅) | |
9 | 7, 8 | 2thd 175 | . . 3 ⊢ (∅ ∈ 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
10 | 6, 9 | jaoi 717 | . 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 105 ∨ wo 709 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ∅c0 3437 ωcom 4604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-int 3860 df-suc 4386 df-iom 4605 |
This theorem is referenced by: nnmord 6537 bj-charfunr 15000 |
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