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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 2046 | . . 3 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅)) | |
2 | eleq2 2101 | . . 3 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅)) | |
3 | 1, 2 | orbi12d 707 | . 2 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅))) |
4 | eqeq1 2046 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅)) | |
5 | eleq2 2101 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦)) | |
6 | 4, 5 | orbi12d 707 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦))) |
7 | eqeq1 2046 | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅)) | |
8 | eleq2 2101 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦)) | |
9 | 7, 8 | orbi12d 707 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦))) |
10 | eqeq1 2046 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
11 | eleq2 2101 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴)) | |
12 | 10, 11 | orbi12d 707 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))) |
13 | eqid 2040 | . . 3 ⊢ ∅ = ∅ | |
14 | 13 | orci 650 | . 2 ⊢ (∅ = ∅ ∨ ∅ ∈ ∅) |
15 | 0ex 3884 | . . . . . . 7 ⊢ ∅ ∈ V | |
16 | 15 | sucid 4154 | . . . . . 6 ⊢ ∅ ∈ suc ∅ |
17 | suceq 4139 | . . . . . 6 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅) | |
18 | 16, 17 | syl5eleqr 2127 | . . . . 5 ⊢ (𝑦 = ∅ → ∅ ∈ suc 𝑦) |
19 | 18 | a1i 9 | . . . 4 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦)) |
20 | sssucid 4152 | . . . . . 6 ⊢ 𝑦 ⊆ suc 𝑦 | |
21 | 20 | a1i 9 | . . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦) |
22 | 21 | sseld 2944 | . . . 4 ⊢ (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦)) |
23 | 19, 22 | jaod 637 | . . 3 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦)) |
24 | olc 632 | . . 3 ⊢ (∅ ∈ suc 𝑦 → (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦)) | |
25 | 23, 24 | syl6 29 | . 2 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦))) |
26 | 3, 6, 9, 12, 14, 25 | finds 4323 | 1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 629 = wceq 1243 ∈ wcel 1393 ⊆ wss 2917 ∅c0 3224 suc csuc 4102 ωcom 4313 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-suc 4108 df-iom 4314 |
This theorem is referenced by: nn0eln0 4341 nnsucsssuc 6071 nntri3or 6072 nnm00 6102 ssfiexmid 6336 diffitest 6344 elni2 6412 enq0tr 6532 |
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