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Mirrors > Home > ILE Home > Th. List > nntri2 | GIF version |
Description: A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.) |
Ref | Expression |
---|---|
nntri2 | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirr 4513 | . . . . 5 ⊢ ¬ 𝐴 ∈ 𝐴 | |
2 | eleq2 2228 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐴 ∈ 𝐵)) | |
3 | 1, 2 | mtbii 664 | . . . 4 ⊢ (𝐴 = 𝐵 → ¬ 𝐴 ∈ 𝐵) |
4 | 3 | con2i 617 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 = 𝐵) |
5 | en2lp 4526 | . . . 4 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) | |
6 | 5 | imnani 681 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ 𝐴) |
7 | ioran 742 | . . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 ∈ 𝐴)) | |
8 | 4, 6, 7 | sylanbrc 414 | . 2 ⊢ (𝐴 ∈ 𝐵 → ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) |
9 | nntri3or 6453 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
10 | 3orass 970 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (𝐴 ∈ 𝐵 ∨ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) | |
11 | 9, 10 | sylib 121 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) |
12 | 11 | orcomd 719 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐴 ∈ 𝐵)) |
13 | 12 | ord 714 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) → 𝐴 ∈ 𝐵)) |
14 | 8, 13 | impbid2 142 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 ∨ w3o 966 = wceq 1342 ∈ wcel 2135 ωcom 4562 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2724 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-uni 3785 df-int 3820 df-tr 4076 df-iord 4339 df-on 4341 df-suc 4344 df-iom 4563 |
This theorem is referenced by: nnaord 6469 nnmord 6477 pitric 7254 |
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