Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4592 | . . . . . 6 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On) | |
2 | 1 | adantl 275 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 ∈ On) |
3 | onelss 4370 | . . . . 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 728 | . 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) |
7 | eqimss 3201 | . . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
8 | 7 | adantl 275 | . . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵) |
9 | 8 | orcd 728 | . 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) |
10 | nnon 4592 | . . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
11 | 10 | adantr 274 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ∈ On) |
12 | onelss 4370 | . . . . 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 729 | . 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ∈ 𝐴) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) |
16 | nntri3or 6469 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
17 | 6, 9, 15, 16 | mpjao3dan 1302 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 703 = wceq 1348 ∈ wcel 2141 ⊆ wss 3121 Oncon0 4346 ωcom 4572 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-uni 3795 df-int 3830 df-tr 4086 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 |
This theorem is referenced by: fientri3 6888 |
Copyright terms: Public domain | W3C validator |