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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 4460 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | |
2 | noel 3306 | . . . . 5 ⊢ ¬ ∅ ∈ ∅ | |
3 | eleq2 2158 | . . . . 5 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ ∅ ∈ ∅)) | |
4 | 2, 3 | mtbiri 638 | . . . 4 ⊢ (𝐴 = ∅ → ¬ ∅ ∈ 𝐴) |
5 | nner 2266 | . . . 4 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≠ ∅) | |
6 | 4, 5 | 2falsed 656 | . . 3 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
7 | id 19 | . . . 4 ⊢ (∅ ∈ 𝐴 → ∅ ∈ 𝐴) | |
8 | ne0i 3308 | . . . 4 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅) | |
9 | 7, 8 | 2thd 174 | . . 3 ⊢ (∅ ∈ 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
10 | 6, 9 | jaoi 674 | . 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 667 = wceq 1296 ∈ wcel 1445 ≠ wne 2262 ∅c0 3302 ωcom 4433 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-uni 3676 df-int 3711 df-suc 4222 df-iom 4434 |
This theorem is referenced by: nnmord 6316 nnnninf 6894 |
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