Theorem maxidlnr 35322

Description: A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.)

Hypotheses

Ref	Expression
maxidlnr.1	⊢ 𝐺 = (1st ‘𝑅)
maxidlnr.2	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
maxidlnr	⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ 𝑋)

Proof of Theorem maxidlnr

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		maxidlnr.1	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
2		maxidlnr.2	. . . 4 ⊢ 𝑋 = ran 𝐺
3	1, 2	ismaxidl 35320	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))))
4	3	biimpa 479	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))))
5	4	simp2d 1139	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ 𝑋)

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140   ⊆ wss 3938  ran crn 5558  ‘cfv 6357  1^st c1st 7689  RingOpscrngo 35174  Idlcidl 35287  MaxIdlcmaxidl 35289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fv 6365  df-maxidl 35292

This theorem is referenced by:  maxidln1  35324