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Mirrors > Home > MPE Home > Th. List > xnn0lenn0nn0 | Structured version Visualization version GIF version |
Description: An extended nonnegative integer which is less than or equal to a nonnegative integer is a nonnegative integer. (Contributed by AV, 24-Nov-2021.) |
Ref | Expression |
---|---|
xnn0lenn0nn0 | ⊢ ((𝑀 ∈ ℕ0* ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxnn0 11957 | . . 3 ⊢ (𝑀 ∈ ℕ0* ↔ (𝑀 ∈ ℕ0 ∨ 𝑀 = +∞)) | |
2 | 2a1 28 | . . . 4 ⊢ (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (𝑀 ≤ 𝑁 → 𝑀 ∈ ℕ0))) | |
3 | breq1 5033 | . . . . . . 7 ⊢ (𝑀 = +∞ → (𝑀 ≤ 𝑁 ↔ +∞ ≤ 𝑁)) | |
4 | 3 | adantr 484 | . . . . . 6 ⊢ ((𝑀 = +∞ ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ +∞ ≤ 𝑁)) |
5 | nn0re 11894 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | 5 | rexrd 10680 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ*) |
7 | xgepnf 12546 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℝ* → (+∞ ≤ 𝑁 ↔ 𝑁 = +∞)) | |
8 | 6, 7 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (+∞ ≤ 𝑁 ↔ 𝑁 = +∞)) |
9 | pnfnre 10671 | . . . . . . . . 9 ⊢ +∞ ∉ ℝ | |
10 | eleq1 2877 | . . . . . . . . . . 11 ⊢ (𝑁 = +∞ → (𝑁 ∈ ℕ0 ↔ +∞ ∈ ℕ0)) | |
11 | nn0re 11894 | . . . . . . . . . . . 12 ⊢ (+∞ ∈ ℕ0 → +∞ ∈ ℝ) | |
12 | elnelall 3104 | . . . . . . . . . . . 12 ⊢ (+∞ ∈ ℝ → (+∞ ∉ ℝ → 𝑀 ∈ ℕ0)) | |
13 | 11, 12 | syl 17 | . . . . . . . . . . 11 ⊢ (+∞ ∈ ℕ0 → (+∞ ∉ ℝ → 𝑀 ∈ ℕ0)) |
14 | 10, 13 | syl6bi 256 | . . . . . . . . . 10 ⊢ (𝑁 = +∞ → (𝑁 ∈ ℕ0 → (+∞ ∉ ℝ → 𝑀 ∈ ℕ0))) |
15 | 14 | com13 88 | . . . . . . . . 9 ⊢ (+∞ ∉ ℝ → (𝑁 ∈ ℕ0 → (𝑁 = +∞ → 𝑀 ∈ ℕ0))) |
16 | 9, 15 | ax-mp 5 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑁 = +∞ → 𝑀 ∈ ℕ0)) |
17 | 8, 16 | sylbid 243 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (+∞ ≤ 𝑁 → 𝑀 ∈ ℕ0)) |
18 | 17 | adantl 485 | . . . . . 6 ⊢ ((𝑀 = +∞ ∧ 𝑁 ∈ ℕ0) → (+∞ ≤ 𝑁 → 𝑀 ∈ ℕ0)) |
19 | 4, 18 | sylbid 243 | . . . . 5 ⊢ ((𝑀 = +∞ ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 → 𝑀 ∈ ℕ0)) |
20 | 19 | ex 416 | . . . 4 ⊢ (𝑀 = +∞ → (𝑁 ∈ ℕ0 → (𝑀 ≤ 𝑁 → 𝑀 ∈ ℕ0))) |
21 | 2, 20 | jaoi 854 | . . 3 ⊢ ((𝑀 ∈ ℕ0 ∨ 𝑀 = +∞) → (𝑁 ∈ ℕ0 → (𝑀 ≤ 𝑁 → 𝑀 ∈ ℕ0))) |
22 | 1, 21 | sylbi 220 | . 2 ⊢ (𝑀 ∈ ℕ0* → (𝑁 ∈ ℕ0 → (𝑀 ≤ 𝑁 → 𝑀 ∈ ℕ0))) |
23 | 22 | 3imp 1108 | 1 ⊢ ((𝑀 ∈ ℕ0* ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 class class class wbr 5030 ℝcr 10525 +∞cpnf 10661 ℝ*cxr 10663 ≤ cle 10665 ℕ0cn0 11885 ℕ0*cxnn0 11955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-i2m1 10594 ax-1ne0 10595 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-nn 11626 df-n0 11886 df-xnn0 11956 |
This theorem is referenced by: xnn0le2is012 12627 |
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