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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 4601 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | |
2 | noel 3418 | . . . . 5 ⊢ ¬ ∅ ∈ ∅ | |
3 | eleq2 2234 | . . . . 5 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ ∅ ∈ ∅)) | |
4 | 2, 3 | mtbiri 670 | . . . 4 ⊢ (𝐴 = ∅ → ¬ ∅ ∈ 𝐴) |
5 | nner 2344 | . . . 4 ⊢ (𝐴 = ∅ → ¬ 𝐴 ≠ ∅) | |
6 | 4, 5 | 2falsed 697 | . . 3 ⊢ (𝐴 = ∅ → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
7 | id 19 | . . . 4 ⊢ (∅ ∈ 𝐴 → ∅ ∈ 𝐴) | |
8 | ne0i 3420 | . . . 4 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅) | |
9 | 7, 8 | 2thd 174 | . . 3 ⊢ (∅ ∈ 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
10 | 6, 9 | jaoi 711 | . 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 703 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ∅c0 3414 ωcom 4572 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-uni 3795 df-int 3830 df-suc 4354 df-iom 4573 |
This theorem is referenced by: nnmord 6493 bj-charfunr 13767 |
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