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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 2146 | . . 3 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅)) | |
2 | eleq2 2203 | . . 3 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅)) | |
3 | 1, 2 | orbi12d 782 | . 2 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅))) |
4 | eqeq1 2146 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅)) | |
5 | eleq2 2203 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦)) | |
6 | 4, 5 | orbi12d 782 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦))) |
7 | eqeq1 2146 | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅)) | |
8 | eleq2 2203 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦)) | |
9 | 7, 8 | orbi12d 782 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦))) |
10 | eqeq1 2146 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
11 | eleq2 2203 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴)) | |
12 | 10, 11 | orbi12d 782 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))) |
13 | eqid 2139 | . . 3 ⊢ ∅ = ∅ | |
14 | 13 | orci 720 | . 2 ⊢ (∅ = ∅ ∨ ∅ ∈ ∅) |
15 | 0ex 4055 | . . . . . . 7 ⊢ ∅ ∈ V | |
16 | 15 | sucid 4339 | . . . . . 6 ⊢ ∅ ∈ suc ∅ |
17 | suceq 4324 | . . . . . 6 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅) | |
18 | 16, 17 | eleqtrrid 2229 | . . . . 5 ⊢ (𝑦 = ∅ → ∅ ∈ suc 𝑦) |
19 | 18 | a1i 9 | . . . 4 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦)) |
20 | sssucid 4337 | . . . . . 6 ⊢ 𝑦 ⊆ suc 𝑦 | |
21 | 20 | a1i 9 | . . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦) |
22 | 21 | sseld 3096 | . . . 4 ⊢ (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦)) |
23 | 19, 22 | jaod 706 | . . 3 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦)) |
24 | olc 700 | . . 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 4514 | 1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ⊆ wss 3071 ∅c0 3363 suc csuc 4287 ωcom 4504 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-suc 4293 df-iom 4505 |
This theorem is referenced by: nn0eln0 4533 nnsucsssuc 6388 nntri3or 6389 nnm00 6425 ssfilem 6769 diffitest 6781 fiintim 6817 enumct 7000 elni2 7122 enq0tr 7242 nninfalllemn 13202 |
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