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Mirrors > Home > ILE Home > Th. List > nntri2or2 | GIF version |
Description: A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.) |
Ref | Expression |
---|---|
nntri2or2 | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 4439 | . . . . . 6 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On) | |
2 | 1 | adantl 272 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 ∈ On) |
3 | onelss 4225 | . . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵)) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵)) |
5 | 4 | imp 123 | . . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ 𝐵) |
6 | 5 | orcd 688 | . 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) |
7 | eqimss 3081 | . . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
8 | 7 | adantl 272 | . . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵) |
9 | 8 | orcd 688 | . 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) |
10 | nnon 4439 | . . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
11 | 10 | adantr 271 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ∈ On) |
12 | onelss 4225 | . . . . 5 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | |
13 | 11, 12 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
14 | 13 | imp 123 | . . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴) |
15 | 14 | olcd 689 | . 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ∈ 𝐴) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) |
16 | nntri3or 6270 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
17 | 6, 9, 15, 16 | mpjao3dan 1244 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 665 = wceq 1290 ∈ wcel 1439 ⊆ wss 3002 Oncon0 4201 ωcom 4420 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-nul 3973 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-iinf 4418 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2624 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-pw 3437 df-sn 3458 df-pr 3459 df-uni 3662 df-int 3697 df-tr 3945 df-iord 4204 df-on 4206 df-suc 4209 df-iom 4421 |
This theorem is referenced by: fientri3 6681 |
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