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Mirrors > Home > ILE Home > Th. List > ringn0 | GIF version |
Description: The class of rings is not empty (it is also inhabited, as shown at ring1 13372). (Contributed by AV, 29-Apr-2019.) |
Ref | Expression |
---|---|
ringn0 | β’ Ring β β |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2755 | . 2 β’ π§ β V | |
2 | eqid 2189 | . . 3 β’ {β¨(Baseβndx), {π§}β©, β¨(+gβndx), {β¨β¨π§, π§β©, π§β©}β©, β¨(.rβndx), {β¨β¨π§, π§β©, π§β©}β©} = {β¨(Baseβndx), {π§}β©, β¨(+gβndx), {β¨β¨π§, π§β©, π§β©}β©, β¨(.rβndx), {β¨β¨π§, π§β©, π§β©}β©} | |
3 | 2 | ring1 13372 | . 2 β’ (π§ β V β {β¨(Baseβndx), {π§}β©, β¨(+gβndx), {β¨β¨π§, π§β©, π§β©}β©, β¨(.rβndx), {β¨β¨π§, π§β©, π§β©}β©} β Ring) |
4 | ne0i 3444 | . 2 β’ ({β¨(Baseβndx), {π§}β©, β¨(+gβndx), {β¨β¨π§, π§β©, π§β©}β©, β¨(.rβndx), {β¨β¨π§, π§β©, π§β©}β©} β Ring β Ring β β ) | |
5 | 1, 3, 4 | mp2b 8 | 1 β’ Ring β β |
Colors of variables: wff set class |
Syntax hints: β wcel 2160 β wne 2360 Vcvv 2752 β c0 3437 {csn 3607 {ctp 3609 β¨cop 3610 βcfv 5231 ndxcnx 12477 Basecbs 12480 +gcplusg 12555 .rcmulr 12556 Ringcrg 13311 |
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 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-addass 7931 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-0id 7937 ax-rnegex 7938 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 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-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-tp 3615 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-inn 8938 df-2 8996 df-3 8997 df-n0 9195 df-z 9272 df-uz 9547 df-fz 10027 df-struct 12482 df-ndx 12483 df-slot 12484 df-base 12486 df-sets 12487 df-plusg 12568 df-mulr 12569 df-0g 12729 df-mgm 12798 df-sgrp 12831 df-mnd 12844 df-grp 12914 df-mgp 13236 df-ring 13313 |
This theorem is referenced by: (None) |
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