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Mirrors > Home > MPE Home > Th. List > Mathboxes > maxidln0 | Structured version Visualization version Unicode version |
Description: A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) |
Ref | Expression |
---|---|
maxidln0.1 | |
maxidln0.2 | |
maxidln0.3 | GId |
maxidln0.4 | GId |
Ref | Expression |
---|---|
maxidln0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | maxidlidl 33840 | . . . . 5 | |
2 | maxidln0.1 | . . . . . 6 | |
3 | maxidln0.3 | . . . . . 6 GId | |
4 | 2, 3 | idl0cl 33817 | . . . . 5 |
5 | 1, 4 | syldan 487 | . . . 4 |
6 | maxidln0.2 | . . . . 5 | |
7 | maxidln0.4 | . . . . 5 GId | |
8 | 6, 7 | maxidln1 33843 | . . . 4 |
9 | nelneq 2725 | . . . 4 | |
10 | 5, 8, 9 | syl2anc 693 | . . 3 |
11 | 10 | neqned 2801 | . 2 |
12 | 11 | necomd 2849 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wceq 1483 wcel 1990 wne 2794 cfv 5888 c1st 7166 c2nd 7167 GIdcgi 27344 crngo 33693 cidl 33806 cmaxidl 33808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-riota 6611 df-ov 6653 df-1st 7168 df-2nd 7169 df-grpo 27347 df-gid 27348 df-ablo 27399 df-ass 33642 df-exid 33644 df-mgmOLD 33648 df-sgrOLD 33660 df-mndo 33666 df-rngo 33694 df-idl 33809 df-maxidl 33811 |
This theorem is referenced by: (None) |
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