![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > le9lt10 | Structured version Visualization version GIF version |
Description: A "decimal digit" (i.e. a nonnegative integer less than or equal to 9) is less then 10. (Contributed by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
le9lt10.c | ⊢ 𝐴 ∈ ℕ0 |
le9lt10.e | ⊢ 𝐴 ≤ 9 |
Ref | Expression |
---|---|
le9lt10 | ⊢ 𝐴 < ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | le9lt10.e | . . 3 ⊢ 𝐴 ≤ 9 | |
2 | le9lt10.c | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
3 | 2 | nn0zi 12584 | . . . 4 ⊢ 𝐴 ∈ ℤ |
4 | 9nn0 12493 | . . . . 5 ⊢ 9 ∈ ℕ0 | |
5 | 4 | nn0zi 12584 | . . . 4 ⊢ 9 ∈ ℤ |
6 | zleltp1 12610 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 9 ∈ ℤ) → (𝐴 ≤ 9 ↔ 𝐴 < (9 + 1))) | |
7 | 3, 5, 6 | mp2an 691 | . . 3 ⊢ (𝐴 ≤ 9 ↔ 𝐴 < (9 + 1)) |
8 | 1, 7 | mpbi 229 | . 2 ⊢ 𝐴 < (9 + 1) |
9 | 9p1e10 12676 | . 2 ⊢ (9 + 1) = ;10 | |
10 | 8, 9 | breqtri 5173 | 1 ⊢ 𝐴 < ;10 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2107 class class class wbr 5148 (class class class)co 7406 0cc0 11107 1c1 11108 + caddc 11110 < clt 11245 ≤ cle 11246 9c9 12271 ℕ0cn0 12469 ℤcz 12555 ;cdc 12674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 |
This theorem is referenced by: declth 12704 decltdi 12713 |
Copyright terms: Public domain | W3C validator |