Theorem maxidlmax 16191

Description: A maximal ideal is a maximal proper ideal.

Hypotheses

Ref	Expression
maxidlnr.1	|- G = (1st` R)
maxidlnr.2	|- X = ran G

Assertion

Ref	Expression
maxidlmax	|- (((R e. Ring /\ M e. (MaxIdl` R)) /\ (I e. (Idl` R) /\ M C_ I)) -> (I = M \/ I = X))

Proof of Theorem maxidlmax

Step	Hyp	Ref	Expression
1		sseq2 2639	. . . . . 6 |- (j = I -> (M C_ j <-> M C_ I))
2		eqeq1 1890	. . . . . . 7 |- (j = I -> (j = M <-> I = M))
3		eqeq1 1890	. . . . . . 7 |- (j = I -> (j = X <-> I = X))
4	2, 3	orbi12d 689	. . . . . 6 |- (j = I -> ((j = M \/ j = X) <-> (I = M \/ I = X)))
5	1, 4	imbi12d 688	. . . . 5 |- (j = I -> ((M C_ j -> (j = M \/ j = X)) <-> (M C_ I -> (I = M \/ I = X))))
6	5	rcla4va 2378	. . . 4 |- ((I e. (Idl` R) /\ A.j e. (Idl` R)(M C_ j -> (j = M \/ j = X))) -> (M C_ I -> (I = M \/ I = X)))
7		maxidlnr.1	. . . . . . 7 |- G = (1st` R)
8		maxidlnr.2	. . . . . . 7 |- X = ran G
9	7, 8	ismaxidl 16188	. . . . . 6 |- (R e. Ring -> (M e. (MaxIdl` R) <-> (M e. (Idl` R) /\ M =/= X /\ A.j e. (Idl` R)(M C_ j -> (j = M \/ j = X)))))
10	9	biimpa 460	. . . . 5 |- ((R e. Ring /\ M e. (MaxIdl` R)) -> (M e. (Idl` R) /\ M =/= X /\ A.j e. (Idl` R)(M C_ j -> (j = M \/ j = X))))
11	10	simp3d 890	. . . 4 |- ((R e. Ring /\ M e. (MaxIdl` R)) -> A.j e. (Idl` R)(M C_ j -> (j = M \/ j = X)))
12	6, 11	sylan2 500	. . 3 |- ((I e. (Idl` R) /\ (R e. Ring /\ M e. (MaxIdl` R))) -> (M C_ I -> (I = M \/ I = X)))
13	12	ancoms 484	. 2 |- (((R e. Ring /\ M e. (MaxIdl` R)) /\ I e. (Idl` R)) -> (M C_ I -> (I = M \/ I = X)))
14	13	impr 422	1 |- (((R e. Ring /\ M e. (MaxIdl` R)) /\ (I e. (Idl` R) /\ M C_ I)) -> (I = M \/ I = X))

Syntax hints:   -> wi 3   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105   C_ wss 2593  ran crn 3987  ` cfv 3998  1stc1st 5018  Ringcring 9463  Idlcidl 16155  MaxIdlcmaxidl 16157

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-maxidl 16160

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