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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 2091 | . . 3 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅)) | |
2 | eleq2 2148 | . . 3 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅)) | |
3 | 1, 2 | orbi12d 740 | . 2 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅))) |
4 | eqeq1 2091 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅)) | |
5 | eleq2 2148 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦)) | |
6 | 4, 5 | orbi12d 740 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦))) |
7 | eqeq1 2091 | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅)) | |
8 | eleq2 2148 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦)) | |
9 | 7, 8 | orbi12d 740 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦))) |
10 | eqeq1 2091 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
11 | eleq2 2148 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴)) | |
12 | 10, 11 | orbi12d 740 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))) |
13 | eqid 2085 | . . 3 ⊢ ∅ = ∅ | |
14 | 13 | orci 683 | . 2 ⊢ (∅ = ∅ ∨ ∅ ∈ ∅) |
15 | 0ex 3934 | . . . . . . 7 ⊢ ∅ ∈ V | |
16 | 15 | sucid 4211 | . . . . . 6 ⊢ ∅ ∈ suc ∅ |
17 | suceq 4196 | . . . . . 6 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅) | |
18 | 16, 17 | syl5eleqr 2174 | . . . . 5 ⊢ (𝑦 = ∅ → ∅ ∈ suc 𝑦) |
19 | 18 | a1i 9 | . . . 4 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦)) |
20 | sssucid 4209 | . . . . . 6 ⊢ 𝑦 ⊆ suc 𝑦 | |
21 | 20 | a1i 9 | . . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦) |
22 | 21 | sseld 3011 | . . . 4 ⊢ (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦)) |
23 | 19, 22 | jaod 670 | . . 3 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦)) |
24 | olc 665 | . . 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 4381 | 1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 662 = wceq 1287 ∈ wcel 1436 ⊆ wss 2986 ∅c0 3272 suc csuc 4159 ωcom 4371 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-iinf 4369 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ral 2360 df-rex 2361 df-v 2616 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-uni 3631 df-int 3666 df-suc 4165 df-iom 4372 |
This theorem is referenced by: nn0eln0 4399 nnsucsssuc 6188 nntri3or 6189 nnm00 6221 ssfilem 6524 diffitest 6536 elni2 6794 enq0tr 6914 nninfalllemn 11254 |
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