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Mirrors > Home > ILE Home > Th. List > nntri1 | GIF version |
Description: A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.) |
Ref | Expression |
---|---|
nntri1 | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssnel 4413 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴) | |
2 | nntri3or 6294 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
3 | df-3or 928 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴)) | |
4 | 3 | biimpi 119 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴)) |
5 | 4 | orcomd 686 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) → (𝐵 ∈ 𝐴 ∨ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
6 | 5 | ord 681 | . . . 4 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) → (¬ 𝐵 ∈ 𝐴 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
7 | 2, 6 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (¬ 𝐵 ∈ 𝐴 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
8 | nnord 4454 | . . . . . . 7 ⊢ (𝐵 ∈ ω → Ord 𝐵) | |
9 | ordelss 4230 | . . . . . . 7 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ 𝐵) | |
10 | 8, 9 | sylan 278 | . . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ 𝐵) |
11 | 10 | ex 114 | . . . . 5 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵)) |
12 | 11 | adantl 272 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵)) |
13 | eqimss 3093 | . . . . 5 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
14 | 13 | a1i 9 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)) |
15 | 12, 14 | jaod 675 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵)) |
16 | 7, 15 | syld 45 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (¬ 𝐵 ∈ 𝐴 → 𝐴 ⊆ 𝐵)) |
17 | 1, 16 | impbid2 142 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 667 ∨ w3o 926 = wceq 1296 ∈ wcel 1445 ⊆ wss 3013 Ord word 4213 ωcom 4433 |
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 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-uni 3676 df-int 3711 df-tr 3959 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 |
This theorem is referenced by: nnsseleq 6302 nnmword 6317 nnawordex 6327 nndomo 6660 nninfalllemn 12607 |
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