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Theorem ssmxidl 33502
Description: Let 𝑅 be a ring, and let 𝐼 be a proper ideal of 𝑅. Then there is a maximal ideal of 𝑅 containing 𝐼. (Contributed by Thierry Arnoux, 10-Apr-2024.)
Hypothesis
Ref Expression
ssmxidl.1 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
ssmxidl ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼𝑚)
Distinct variable groups:   𝐵,𝑚   𝑚,𝐼   𝑅,𝑚

Proof of Theorem ssmxidl
Dummy variables 𝑗 𝑝 𝑧 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neeq1 3003 . . . . . 6 (𝑝 = 𝐼 → (𝑝𝐵𝐼𝐵))
2 sseq2 4010 . . . . . 6 (𝑝 = 𝐼 → (𝐼𝑝𝐼𝐼))
31, 2anbi12d 632 . . . . 5 (𝑝 = 𝐼 → ((𝑝𝐵𝐼𝑝) ↔ (𝐼𝐵𝐼𝐼)))
4 simp2 1138 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → 𝐼 ∈ (LIdeal‘𝑅))
5 simp3 1139 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → 𝐼𝐵)
6 ssidd 4007 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → 𝐼𝐼)
75, 6jca 511 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → (𝐼𝐵𝐼𝐼))
83, 4, 7elrabd 3694 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → 𝐼 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)})
98ne0d 4342 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ≠ ∅)
10 ssmxidl.1 . . . . . 6 𝐵 = (Base‘𝑅)
11 eqid 2737 . . . . . 6 {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} = {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}
12 simpl1 1192 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → 𝑅 ∈ Ring)
13 simpl2 1193 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → 𝐼 ∈ (LIdeal‘𝑅))
14 simpl3 1194 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → 𝐼𝐵)
15 simpr1 1195 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → 𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)})
16 simpr2 1196 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → 𝑧 ≠ ∅)
17 simpr3 1197 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → [] Or 𝑧)
1810, 11, 12, 13, 14, 15, 16, 17ssmxidllem 33501 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → 𝑧 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)})
1918ex 412 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ((𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}))
2019alrimiv 1927 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ∀𝑧((𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}))
21 fvex 6919 . . . . 5 (LIdeal‘𝑅) ∈ V
2221rabex 5339 . . . 4 {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∈ V
2322zornn0 10548 . . 3 (({𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)})) → ∃𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗)
249, 20, 23syl2anc 584 . 2 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ∃𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗)
25 neeq1 3003 . . . . . . . 8 (𝑝 = 𝑚 → (𝑝𝐵𝑚𝐵))
26 sseq2 4010 . . . . . . . 8 (𝑝 = 𝑚 → (𝐼𝑝𝐼𝑚))
2725, 26anbi12d 632 . . . . . . 7 (𝑝 = 𝑚 → ((𝑝𝐵𝐼𝑝) ↔ (𝑚𝐵𝐼𝑚)))
2827elrab 3692 . . . . . 6 (𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ↔ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚)))
2928anbi2i 623 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ 𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}) ↔ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))))
30 simpll1 1213 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → 𝑅 ∈ Ring)
31 simplrl 777 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → 𝑚 ∈ (LIdeal‘𝑅))
32 simplr 769 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚)))
3332simprld 772 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → 𝑚𝐵)
34 psseq2 4091 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → (𝑚𝑗𝑚𝑘))
3534notbid 318 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (¬ 𝑚𝑗 ↔ ¬ 𝑚𝑘))
36 simp-4r 784 . . . . . . . . . . . . . . 15 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗)
37 neeq1 3003 . . . . . . . . . . . . . . . . 17 (𝑝 = 𝑘 → (𝑝𝐵𝑘𝐵))
38 sseq2 4010 . . . . . . . . . . . . . . . . 17 (𝑝 = 𝑘 → (𝐼𝑝𝐼𝑘))
3937, 38anbi12d 632 . . . . . . . . . . . . . . . 16 (𝑝 = 𝑘 → ((𝑝𝐵𝐼𝑝) ↔ (𝑘𝐵𝐼𝑘)))
40 simpllr 776 . . . . . . . . . . . . . . . 16 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝑘 ∈ (LIdeal‘𝑅))
41 simpr 484 . . . . . . . . . . . . . . . . . 18 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → ¬ 𝑘 = 𝐵)
4241neqned 2947 . . . . . . . . . . . . . . . . 17 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝑘𝐵)
43 simp-5r 786 . . . . . . . . . . . . . . . . . . 19 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚)))
4443simprrd 774 . . . . . . . . . . . . . . . . . 18 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝐼𝑚)
45 simplr 769 . . . . . . . . . . . . . . . . . 18 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝑚𝑘)
4644, 45sstrd 3994 . . . . . . . . . . . . . . . . 17 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝐼𝑘)
4742, 46jca 511 . . . . . . . . . . . . . . . 16 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → (𝑘𝐵𝐼𝑘))
4839, 40, 47elrabd 3694 . . . . . . . . . . . . . . 15 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝑘 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)})
4935, 36, 48rspcdva 3623 . . . . . . . . . . . . . 14 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → ¬ 𝑚𝑘)
50 npss 4113 . . . . . . . . . . . . . . 15 𝑚𝑘 ↔ (𝑚𝑘𝑚 = 𝑘))
5150biimpi 216 . . . . . . . . . . . . . 14 𝑚𝑘 → (𝑚𝑘𝑚 = 𝑘))
5249, 45, 51sylc 65 . . . . . . . . . . . . 13 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝑚 = 𝑘)
5352equcomd 2018 . . . . . . . . . . . 12 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝑘 = 𝑚)
5453ex 412 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) → (¬ 𝑘 = 𝐵𝑘 = 𝑚))
5554orrd 864 . . . . . . . . . 10 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) → (𝑘 = 𝐵𝑘 = 𝑚))
5655orcomd 872 . . . . . . . . 9 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) → (𝑘 = 𝑚𝑘 = 𝐵))
5756ex 412 . . . . . . . 8 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → (𝑚𝑘 → (𝑘 = 𝑚𝑘 = 𝐵)))
5857ralrimiva 3146 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → ∀𝑘 ∈ (LIdeal‘𝑅)(𝑚𝑘 → (𝑘 = 𝑚𝑘 = 𝐵)))
5910ismxidl 33490 . . . . . . . 8 (𝑅 ∈ Ring → (𝑚 ∈ (MaxIdeal‘𝑅) ↔ (𝑚 ∈ (LIdeal‘𝑅) ∧ 𝑚𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑚𝑘 → (𝑘 = 𝑚𝑘 = 𝐵)))))
6059biimpar 477 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ 𝑚𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑚𝑘 → (𝑘 = 𝑚𝑘 = 𝐵)))) → 𝑚 ∈ (MaxIdeal‘𝑅))
6130, 31, 33, 58, 60syl13anc 1374 . . . . . 6 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → 𝑚 ∈ (MaxIdeal‘𝑅))
6232simprrd 774 . . . . . 6 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → 𝐼𝑚)
6361, 62jca 511 . . . . 5 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → (𝑚 ∈ (MaxIdeal‘𝑅) ∧ 𝐼𝑚))
6429, 63sylanb 581 . . . 4 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ 𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → (𝑚 ∈ (MaxIdeal‘𝑅) ∧ 𝐼𝑚))
6564expl 457 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ((𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → (𝑚 ∈ (MaxIdeal‘𝑅) ∧ 𝐼𝑚)))
6665reximdv2 3164 . 2 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → (∃𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗 → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼𝑚))
6724, 66mpd 15 1 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼𝑚)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848  w3a 1087  wal 1538   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  wss 3951  wpss 3952  c0 4333   cuni 4907   Or wor 5591  cfv 6561   [] crpss 7742  Basecbs 17247  Ringcrg 20230  LIdealclidl 21216  MaxIdealcmxidl 33487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rpss 7743  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-subrg 20570  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-mxidl 33488
This theorem is referenced by:  drngmxidlr  33506  krull  33507  zarcls1  33868  zarclssn  33872
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