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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 2196 | . . 3 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅)) | |
2 | eleq2 2253 | . . 3 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅)) | |
3 | 1, 2 | orbi12d 794 | . 2 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅))) |
4 | eqeq1 2196 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅)) | |
5 | eleq2 2253 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦)) | |
6 | 4, 5 | orbi12d 794 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦))) |
7 | eqeq1 2196 | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅)) | |
8 | eleq2 2253 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦)) | |
9 | 7, 8 | orbi12d 794 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦))) |
10 | eqeq1 2196 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
11 | eleq2 2253 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴)) | |
12 | 10, 11 | orbi12d 794 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))) |
13 | eqid 2189 | . . 3 ⊢ ∅ = ∅ | |
14 | 13 | orci 732 | . 2 ⊢ (∅ = ∅ ∨ ∅ ∈ ∅) |
15 | 0ex 4145 | . . . . . . 7 ⊢ ∅ ∈ V | |
16 | 15 | sucid 4435 | . . . . . 6 ⊢ ∅ ∈ suc ∅ |
17 | suceq 4420 | . . . . . 6 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅) | |
18 | 16, 17 | eleqtrrid 2279 | . . . . 5 ⊢ (𝑦 = ∅ → ∅ ∈ suc 𝑦) |
19 | 18 | a1i 9 | . . . 4 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦)) |
20 | sssucid 4433 | . . . . . 6 ⊢ 𝑦 ⊆ suc 𝑦 | |
21 | 20 | a1i 9 | . . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦) |
22 | 21 | sseld 3169 | . . . 4 ⊢ (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦)) |
23 | 19, 22 | jaod 718 | . . 3 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦)) |
24 | olc 712 | . . 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 4617 | 1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 709 = wceq 1364 ∈ wcel 2160 ⊆ wss 3144 ∅c0 3437 suc csuc 4383 ωcom 4607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-int 3860 df-suc 4389 df-iom 4608 |
This theorem is referenced by: nn0eln0 4637 nnsucsssuc 6516 nntri3or 6517 nnm00 6554 ssfilem 6902 diffitest 6914 fiintim 6956 enumct 7143 nnnninfeq 7155 elni2 7342 enq0tr 7462 bj-charfunr 15015 |
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