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Mirrors > Home > MPE Home > Th. List > abln0 | Structured version Visualization version GIF version |
Description: Abelian groups (and therefore also groups and monoids) exist. (Contributed by AV, 29-Apr-2019.) |
Ref | Expression |
---|---|
abln0 | ⊢ Abel ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3354 | . 2 ⊢ 𝑖 ∈ V | |
2 | eqid 2771 | . . 3 ⊢ {〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉} = {〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉} | |
3 | 2 | abl1 18477 | . 2 ⊢ (𝑖 ∈ V → {〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉} ∈ Abel) |
4 | ne0i 4070 | . 2 ⊢ ({〈(Base‘ndx), {𝑖}〉, 〈(+g‘ndx), {〈〈𝑖, 𝑖〉, 𝑖〉}〉} ∈ Abel → Abel ≠ ∅) | |
5 | 1, 3, 4 | mp2b 10 | 1 ⊢ Abel ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2145 ≠ wne 2943 Vcvv 3351 ∅c0 4064 {csn 4317 {cpr 4319 〈cop 4323 ‘cfv 6032 ndxcnx 16062 Basecbs 16065 +gcplusg 16150 Abelcabl 18402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 ax-pre-mulgt0 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-1st 7316 df-2nd 7317 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-1o 7714 df-oadd 7718 df-er 7897 df-en 8111 df-dom 8112 df-sdom 8113 df-fin 8114 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-le 10283 df-sub 10471 df-neg 10472 df-nn 11224 df-2 11282 df-n0 11496 df-z 11581 df-uz 11890 df-fz 12535 df-struct 16067 df-ndx 16068 df-slot 16069 df-base 16071 df-plusg 16163 df-0g 16311 df-mgm 17451 df-sgrp 17493 df-mnd 17504 df-grp 17634 df-cmn 18403 df-abl 18404 |
This theorem is referenced by: (None) |
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