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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 9302 | . . . 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 9302 | . . . 4 ⊢ ((((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 ∈ ℕ0) ∧ 𝐴 ∈ ℕ0) → 𝐵 ∈ ℤ) |
5 | zletric 9226 | . . . 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 9173 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ0* → 𝐵 ∈ ℝ*) | |
8 | pnfge 9716 | . . . . . . 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 4004 | . . . 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 9170 | . . . . 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 9173 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*) | |
19 | 18 | ad2antrr 480 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 = +∞) → 𝐴 ∈ ℝ*) |
20 | pnfge 9716 | . . . . 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 4004 | . . 3 ⊢ (((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 = +∞) → 𝐴 ≤ 𝐵) |
24 | 23 | orcd 723 | . 2 ⊢ (((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ 𝐵 = +∞) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
25 | elxnn0 9170 | . . . 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 3976 +∞cpnf 7921 ℝ*cxr 7923 ≤ cle 7925 ℕ0cn0 9105 ℕ0*cxnn0 9168 ℤcz 9182 |
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 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
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 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-xnn0 9169 df-z 9183 |
This theorem is referenced by: pcgcd 12237 |
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