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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 4283 | . 2 ⊢ (A ∈ 𝜔 → (A = ∅ ∨ ∅ ∈ A)) | |
2 | noel 3222 | . . . . 5 ⊢ ¬ ∅ ∈ ∅ | |
3 | eleq2 2098 | . . . . 5 ⊢ (A = ∅ → (∅ ∈ A ↔ ∅ ∈ ∅)) | |
4 | 2, 3 | mtbiri 599 | . . . 4 ⊢ (A = ∅ → ¬ ∅ ∈ A) |
5 | nner 2207 | . . . 4 ⊢ (A = ∅ → ¬ A ≠ ∅) | |
6 | 4, 5 | 2falsed 617 | . . 3 ⊢ (A = ∅ → (∅ ∈ A ↔ A ≠ ∅)) |
7 | id 19 | . . . 4 ⊢ (∅ ∈ A → ∅ ∈ A) | |
8 | ne0i 3224 | . . . 4 ⊢ (∅ ∈ A → A ≠ ∅) | |
9 | 7, 8 | 2thd 164 | . . 3 ⊢ (∅ ∈ A → (∅ ∈ A ↔ A ≠ ∅)) |
10 | 6, 9 | jaoi 635 | . 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 98 ∨ wo 628 = wceq 1242 ∈ wcel 1390 ≠ wne 2201 ∅c0 3218 𝜔com 4256 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-uni 3572 df-int 3607 df-suc 4074 df-iom 4257 |
This theorem is referenced by: nnmord 6026 |
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