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Mirrors > Home > ILE Home > Th. List > submid | Unicode version |
Description: Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
Ref | Expression |
---|---|
submss.b |
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Ref | Expression |
---|---|
submid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssidd 3200 |
. 2
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2 | submss.b |
. . 3
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3 | eqid 2193 |
. . 3
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4 | 2, 3 | mndidcl 13001 |
. 2
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5 | eqid 2193 |
. . . . 5
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6 | 2, 5 | mndcl 12994 |
. . . 4
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7 | 6 | 3expb 1206 |
. . 3
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8 | 7 | ralrimivva 2576 |
. 2
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9 | 2, 3, 5 | issubm 13034 |
. 2
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10 | 1, 4, 8, 9 | mpbir3and 1182 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-cnex 7953 ax-resscn 7954 ax-1re 7956 ax-addrcl 7959 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4322 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-iota 5207 df-fun 5248 df-fn 5249 df-fv 5254 df-riota 5865 df-ov 5913 df-inn 8973 df-2 9031 df-ndx 12611 df-slot 12612 df-base 12614 df-plusg 12698 df-0g 12859 df-mgm 12929 df-sgrp 12975 df-mnd 12988 df-submnd 13022 |
This theorem is referenced by: gsumwcl 13059 |
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