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Metamath Proof Explorer

Theorem mxidlmaxv

Description: An ideal I strictly containing a maximal ideal M is the whole ring B . (Contributed by Thierry Arnoux, 9-Mar-2025)

Ref Expression
Hypotheses mxidlmaxv.1 
|- B = ( Base ` R )
mxidlmaxv.2 
|- ( ph -> R e. Ring )
mxidlmaxv.3 
|- ( ph -> M e. ( MaxIdeal ` R ) )
mxidlmaxv.4 
|- ( ph -> I e. ( LIdeal ` R ) )
mxidlmaxv.5 
|- ( ph -> M C_ I )
mxidlmaxv.6 
|- ( ph -> X e. ( I \ M ) )
Assertion mxidlmaxv 
|- ( ph -> I = B )

Proof

Step Hyp Ref Expression
1 mxidlmaxv.1 
 |-  B = ( Base ` R )
2 mxidlmaxv.2 
 |-  ( ph -> R e. Ring )
3 mxidlmaxv.3 
 |-  ( ph -> M e. ( MaxIdeal ` R ) )
4 mxidlmaxv.4 
 |-  ( ph -> I e. ( LIdeal ` R ) )
5 mxidlmaxv.5 
 |-  ( ph -> M C_ I )
6 mxidlmaxv.6 
 |-  ( ph -> X e. ( I \ M ) )
7 1 mxidlmax 
 |-  ( ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) /\ ( I e. ( LIdeal ` R ) /\ M C_ I ) ) -> ( I = M \/ I = B ) )
8 2 3 4 5 7 syl22anc 
 |-  ( ph -> ( I = M \/ I = B ) )
9 6 eldifad 
 |-  ( ph -> X e. I )
10 6 eldifbd 
 |-  ( ph -> -. X e. M )
11 nelne1 
 |-  ( ( X e. I /\ -. X e. M ) -> I =/= M )
12 9 10 11 syl2anc 
 |-  ( ph -> I =/= M )
13 12 neneqd 
 |-  ( ph -> -. I = M )
14 8 13 orcnd 
 |-  ( ph -> I = B )