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Mirrors > Home > MPE Home > Th. List > cntz2ss | Structured version Visualization version Unicode version |
Description: Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.) |
Ref | Expression |
---|---|
cntzrec.b | |
cntzrec.z | Cntz |
Ref | Expression |
---|---|
cntz2ss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . . 6 | |
2 | cntzrec.z | . . . . . 6 Cntz | |
3 | 1, 2 | cntzi 17762 | . . . . 5 |
4 | 3 | ralrimiva 2966 | . . . 4 |
5 | ssralv 3666 | . . . . 5 | |
6 | 5 | adantl 482 | . . . 4 |
7 | 4, 6 | syl5 34 | . . 3 |
8 | 7 | ralrimiv 2965 | . 2 |
9 | cntzrec.b | . . . 4 | |
10 | 9, 2 | cntzssv 17761 | . . 3 |
11 | sstr 3611 | . . . 4 | |
12 | 11 | ancoms 469 | . . 3 |
13 | 9, 1, 2 | sscntz 17759 | . . 3 |
14 | 10, 12, 13 | sylancr 695 | . 2 |
15 | 8, 14 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 wss 3574 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 Cntzccntz 17748 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
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-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-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-cntz 17750 |
This theorem is referenced by: cntzidss 17770 gsumzadd 18322 dprdfadd 18419 dprdss 18428 dprd2da 18441 dmdprdsplit2lem 18444 cntzsdrg 37772 |
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