Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > le9lt10 | 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 9100 | . . . 4 ⊢ 𝐴 ∈ ℤ |
4 | 9nn0 9025 | . . . . 5 ⊢ 9 ∈ ℕ0 | |
5 | 4 | nn0zi 9100 | . . . 4 ⊢ 9 ∈ ℤ |
6 | zleltp1 9133 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 9 ∈ ℤ) → (𝐴 ≤ 9 ↔ 𝐴 < (9 + 1))) | |
7 | 3, 5, 6 | mp2an 423 | . . 3 ⊢ (𝐴 ≤ 9 ↔ 𝐴 < (9 + 1)) |
8 | 1, 7 | mpbi 144 | . 2 ⊢ 𝐴 < (9 + 1) |
9 | 9p1e10 9208 | . 2 ⊢ (9 + 1) = ;10 | |
10 | 8, 9 | breqtri 3961 | 1 ⊢ 𝐴 < ;10 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1481 class class class wbr 3937 (class class class)co 5782 0cc0 7644 1c1 7645 + caddc 7647 < clt 7824 ≤ cle 7825 9c9 8802 ℕ0cn0 9001 ℤcz 9078 ;cdc 9206 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 df-n0 9002 df-z 9079 df-dec 9207 |
This theorem is referenced by: declth 9235 decltdi 9244 |
Copyright terms: Public domain | W3C validator |