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Mirrors > Home > ILE Home > Th. List > 0elnn | GIF version |
Description: A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.) |
Ref | Expression |
---|---|
0elnn | ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2106 | . . 3 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅)) | |
2 | eleq2 2163 | . . 3 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅)) | |
3 | 1, 2 | orbi12d 748 | . 2 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅))) |
4 | eqeq1 2106 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅)) | |
5 | eleq2 2163 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦)) | |
6 | 4, 5 | orbi12d 748 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦))) |
7 | eqeq1 2106 | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅)) | |
8 | eleq2 2163 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦)) | |
9 | 7, 8 | orbi12d 748 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦))) |
10 | eqeq1 2106 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
11 | eleq2 2163 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴)) | |
12 | 10, 11 | orbi12d 748 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))) |
13 | eqid 2100 | . . 3 ⊢ ∅ = ∅ | |
14 | 13 | orci 691 | . 2 ⊢ (∅ = ∅ ∨ ∅ ∈ ∅) |
15 | 0ex 3995 | . . . . . . 7 ⊢ ∅ ∈ V | |
16 | 15 | sucid 4277 | . . . . . 6 ⊢ ∅ ∈ suc ∅ |
17 | suceq 4262 | . . . . . 6 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅) | |
18 | 16, 17 | syl5eleqr 2189 | . . . . 5 ⊢ (𝑦 = ∅ → ∅ ∈ suc 𝑦) |
19 | 18 | a1i 9 | . . . 4 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦)) |
20 | sssucid 4275 | . . . . . 6 ⊢ 𝑦 ⊆ suc 𝑦 | |
21 | 20 | a1i 9 | . . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦) |
22 | 21 | sseld 3046 | . . . 4 ⊢ (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦)) |
23 | 19, 22 | jaod 678 | . . 3 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦)) |
24 | olc 673 | . . 3 ⊢ (∅ ∈ suc 𝑦 → (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦)) | |
25 | 23, 24 | syl6 33 | . 2 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦))) |
26 | 3, 6, 9, 12, 14, 25 | finds 4452 | 1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 670 = wceq 1299 ∈ wcel 1448 ⊆ wss 3021 ∅c0 3310 suc csuc 4225 ωcom 4442 |
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 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-uni 3684 df-int 3719 df-suc 4231 df-iom 4443 |
This theorem is referenced by: nn0eln0 4471 nnsucsssuc 6318 nntri3or 6319 nnm00 6355 ssfilem 6698 diffitest 6710 fiintim 6746 enumct 6914 elni2 7023 enq0tr 7143 nninfalllemn 12786 |
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