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Mirrors > Home > ILE Home > Th. List > n2dvds3 | GIF version |
Description: 2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.) |
Ref | Expression |
---|---|
n2dvds3 | ⊢ ¬ 2 ∥ 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 9310 | . . . 4 ⊢ 2 ∈ ℤ | |
2 | iddvds 11842 | . . . 4 ⊢ (2 ∈ ℤ → 2 ∥ 2) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 2 ∥ 2 |
4 | 3m1e2 9068 | . . 3 ⊢ (3 − 1) = 2 | |
5 | 3, 4 | breqtrri 4045 | . 2 ⊢ 2 ∥ (3 − 1) |
6 | 3z 9311 | . . 3 ⊢ 3 ∈ ℤ | |
7 | oddm1even 11911 | . . 3 ⊢ (3 ∈ ℤ → (¬ 2 ∥ 3 ↔ 2 ∥ (3 − 1))) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ (¬ 2 ∥ 3 ↔ 2 ∥ (3 − 1)) |
9 | 5, 8 | mpbir 146 | 1 ⊢ ¬ 2 ∥ 3 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 105 ∈ wcel 2160 class class class wbr 4018 (class class class)co 5895 1c1 7841 − cmin 8157 2c2 8999 3c3 9000 ℤcz 9282 ∥ cdvds 11825 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 ax-pre-mulext 7958 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-div 8659 df-inn 8949 df-2 9007 df-3 9008 df-n0 9206 df-z 9283 df-dvds 11826 |
This theorem is referenced by: 2lgsoddprmlem3 14912 |
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