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Mirrors > Home > ILE Home > Th. List > xnn0letri | GIF version |
Description: Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.) |
Ref | Expression |
---|---|
xnn0letri | ⊢ ((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . 5 ⊢ ((((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 ∈ ℕ0) ∧ 𝐴 ∈ ℕ0) → 𝐴 ∈ ℕ0) | |
2 | 1 | nn0zd 9305 | . . . 4 ⊢ ((((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 ∈ ℕ0) ∧ 𝐴 ∈ ℕ0) → 𝐴 ∈ ℤ) |
3 | simplr 520 | . . . . 5 ⊢ ((((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 ∈ ℕ0) ∧ 𝐴 ∈ ℕ0) → 𝐵 ∈ ℕ0) | |
4 | 3 | nn0zd 9305 | . . . 4 ⊢ ((((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 ∈ ℕ0) ∧ 𝐴 ∈ ℕ0) → 𝐵 ∈ ℤ) |
5 | zletric 9229 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | |
6 | 2, 4, 5 | syl2anc 409 | . . 3 ⊢ ((((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 ∈ ℕ0) ∧ 𝐴 ∈ ℕ0) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
7 | xnn0xr 9176 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ0* → 𝐵 ∈ ℝ*) | |
8 | pnfge 9719 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ +∞) | |
9 | 7, 8 | syl 14 | . . . . . 6 ⊢ (𝐵 ∈ ℕ0* → 𝐵 ≤ +∞) |
10 | 9 | ad3antlr 485 | . . . . 5 ⊢ ((((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 ∈ ℕ0) ∧ 𝐴 = +∞) → 𝐵 ≤ +∞) |
11 | simpr 109 | . . . . 5 ⊢ ((((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 ∈ ℕ0) ∧ 𝐴 = +∞) → 𝐴 = +∞) | |
12 | 10, 11 | breqtrrd 4007 | . . . 4 ⊢ ((((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 ∈ ℕ0) ∧ 𝐴 = +∞) → 𝐵 ≤ 𝐴) |
13 | 12 | olcd 724 | . . 3 ⊢ ((((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 ∈ ℕ0) ∧ 𝐴 = +∞) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
14 | elxnn0 9173 | . . . . 5 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) | |
15 | 14 | biimpi 119 | . . . 4 ⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) |
16 | 15 | ad2antrr 480 | . . 3 ⊢ (((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 ∈ ℕ0) → (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) |
17 | 6, 13, 16 | mpjaodan 788 | . 2 ⊢ (((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 ∈ ℕ0) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
18 | xnn0xr 9176 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*) | |
19 | 18 | ad2antrr 480 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 = +∞) → 𝐴 ∈ ℝ*) |
20 | pnfge 9719 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) | |
21 | 19, 20 | syl 14 | . . . 4 ⊢ (((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 = +∞) → 𝐴 ≤ +∞) |
22 | simpr 109 | . . . 4 ⊢ (((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 = +∞) → 𝐵 = +∞) | |
23 | 21, 22 | breqtrrd 4007 | . . 3 ⊢ (((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 = +∞) → 𝐴 ≤ 𝐵) |
24 | 23 | orcd 723 | . 2 ⊢ (((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 = +∞) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
25 | elxnn0 9173 | . . . 4 ⊢ (𝐵 ∈ ℕ0* ↔ (𝐵 ∈ ℕ0 ∨ 𝐵 = +∞)) | |
26 | 25 | biimpi 119 | . . 3 ⊢ (𝐵 ∈ ℕ0* → (𝐵 ∈ ℕ0 ∨ 𝐵 = +∞)) |
27 | 26 | adantl 275 | . 2 ⊢ ((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → (𝐵 ∈ ℕ0 ∨ 𝐵 = +∞)) |
28 | 17, 24, 27 | mpjaodan 788 | 1 ⊢ ((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 698 = wceq 1342 ∈ wcel 2135 class class class wbr 3979 +∞cpnf 7924 ℝ*cxr 7926 ≤ cle 7928 ℕ0cn0 9108 ℕ0*cxnn0 9171 ℤcz 9185 |
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 4097 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-addcom 7847 ax-addass 7849 ax-distr 7851 ax-i2m1 7852 ax-0lt1 7853 ax-0id 7855 ax-rnegex 7856 ax-cnre 7858 ax-pre-ltirr 7859 ax-pre-ltwlin 7860 ax-pre-lttrn 7861 ax-pre-ltadd 7863 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2726 df-sbc 2950 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-br 3980 df-opab 4041 df-id 4268 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-iota 5150 df-fun 5187 df-fv 5193 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 df-le 7933 df-sub 8065 df-neg 8066 df-inn 8852 df-n0 9109 df-xnn0 9172 df-z 9186 |
This theorem is referenced by: pcgcd 12254 |
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