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| Mirrors > Home > MPE Home > Th. List > lidlunin0 | Structured version Visualization version GIF version | ||
| Description: The union of a nonempty subset of ideals in a ring is nonempty. (Contributed by AV, 28-Jun-2026.) |
| Ref | Expression |
|---|---|
| lidlunin0 | ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ∪ 𝐶 ≠ ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | n0 4315 | . . . . . . 7 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐶) | |
| 2 | simpl 487 | . . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → 𝑅 ∈ Ring) | |
| 3 | ssel 3939 | . . . . . . . . . . . . 13 ⊢ (𝐶 ⊆ (LIdeal‘𝑅) → (𝑦 ∈ 𝐶 → 𝑦 ∈ (LIdeal‘𝑅))) | |
| 4 | 3 | adantl 486 | . . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → (𝑦 ∈ 𝐶 → 𝑦 ∈ (LIdeal‘𝑅))) |
| 5 | 4 | imp 411 | . . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ (LIdeal‘𝑅)) |
| 6 | eqid 2770 | . . . . . . . . . . . 12 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
| 7 | eqid 2770 | . . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 8 | 6, 7 | lidl0cl 21328 | . . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (LIdeal‘𝑅)) → (0g‘𝑅) ∈ 𝑦) |
| 9 | 2, 5, 8 | syl2an2r 697 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ 𝑦 ∈ 𝐶) → (0g‘𝑅) ∈ 𝑦) |
| 10 | simpr 489 | . . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ 𝐶) | |
| 11 | 9, 10 | jca 520 | . . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ 𝑦 ∈ 𝐶) → ((0g‘𝑅) ∈ 𝑦 ∧ 𝑦 ∈ 𝐶)) |
| 12 | 11 | ex 417 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → (𝑦 ∈ 𝐶 → ((0g‘𝑅) ∈ 𝑦 ∧ 𝑦 ∈ 𝐶))) |
| 13 | 12 | eximdv 1945 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → (∃𝑦 𝑦 ∈ 𝐶 → ∃𝑦((0g‘𝑅) ∈ 𝑦 ∧ 𝑦 ∈ 𝐶))) |
| 14 | 1, 13 | biimtrid 245 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → (𝐶 ≠ ∅ → ∃𝑦((0g‘𝑅) ∈ 𝑦 ∧ 𝑦 ∈ 𝐶))) |
| 15 | 14 | ex 417 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐶 ⊆ (LIdeal‘𝑅) → (𝐶 ≠ ∅ → ∃𝑦((0g‘𝑅) ∈ 𝑦 ∧ 𝑦 ∈ 𝐶)))) |
| 16 | 15 | com23 87 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝐶 ≠ ∅ → (𝐶 ⊆ (LIdeal‘𝑅) → ∃𝑦((0g‘𝑅) ∈ 𝑦 ∧ 𝑦 ∈ 𝐶)))) |
| 17 | 16 | 3imp 1126 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ∃𝑦((0g‘𝑅) ∈ 𝑦 ∧ 𝑦 ∈ 𝐶)) |
| 18 | eluni 4880 | . . 3 ⊢ ((0g‘𝑅) ∈ ∪ 𝐶 ↔ ∃𝑦((0g‘𝑅) ∈ 𝑦 ∧ 𝑦 ∈ 𝐶)) | |
| 19 | 17, 18 | sylibr 237 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → (0g‘𝑅) ∈ ∪ 𝐶) |
| 20 | 19 | ne0d 4303 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ∪ 𝐶 ≠ ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 ∃wex 1807 ∈ wcel 2150 ≠ wne 2965 ⊆ wss 3913 ∅c0 4294 ∪ cuni 4877 ‘cfv 6540 0gc0g 17495 Ringcrg 20318 LIdealclidl 21313 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2152 ax-9 2160 ax-10 2183 ax-11 2199 ax-12 2220 ax-ext 2742 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5340 ax-pr 5408 ax-un 7736 ax-cnex 11159 ax-resscn 11160 ax-1cn 11161 ax-icn 11162 ax-addcl 11163 ax-addrcl 11164 ax-mulcl 11165 ax-mulrcl 11166 ax-mulcom 11167 ax-addass 11168 ax-mulass 11169 ax-distr 11170 ax-i2m1 11171 ax-1ne0 11172 ax-1rid 11173 ax-rnegex 11174 ax-rrecex 11175 ax-cnre 11176 ax-pre-lttri 11177 ax-pre-lttrn 11178 ax-pre-ltadd 11179 ax-pre-mulgt0 11180 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2099 df-mo 2574 df-eu 2604 df-clab 2749 df-cleq 2762 df-clel 2845 df-nfc 2919 df-ne 2966 df-nel 3072 df-ral 3087 df-rex 3097 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3464 df-sbc 3753 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5560 df-eprel 5565 df-po 5573 df-so 5574 df-fr 5618 df-we 5620 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-er 8697 df-en 8947 df-dom 8948 df-sdom 8949 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11446 df-neg 11447 df-nn 12237 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-sets 17227 df-slot 17245 df-ndx 17257 df-base 17273 df-ress 17294 df-plusg 17326 df-mulr 17327 df-sca 17329 df-vsca 17330 df-ip 17331 df-0g 17497 df-mgm 18701 df-sgrp 18780 df-mnd 18796 df-grp 19006 df-minusg 19007 df-sbg 19008 df-subg 19192 df-mgp 20220 df-ur 20267 df-ring 20320 df-subrg 20658 df-lmod 20966 df-lss 21036 df-sra 21277 df-rgmod 21278 df-lidl 21315 |
| This theorem is referenced by: unichnlidl 21345 |
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