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Mirrors > Home > ILE Home > Th. List > mgmidcl | Unicode version |
Description: The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
ismgmid.b |
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ismgmid.o |
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ismgmid.p |
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mgmidcl.e |
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Ref | Expression |
---|---|
mgmidcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2187 |
. . 3
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2 | ismgmid.b |
. . . 4
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3 | ismgmid.o |
. . . 4
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4 | ismgmid.p |
. . . 4
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5 | mgmidcl.e |
. . . 4
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6 | 2, 3, 4, 5 | ismgmid 12814 |
. . 3
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7 | 1, 6 | mpbiri 168 |
. 2
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8 | 7 | simpld 112 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-cnex 7915 ax-resscn 7916 ax-1re 7918 ax-addrcl 7921 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-iota 5190 df-fun 5230 df-fn 5231 df-fv 5236 df-riota 5844 df-ov 5891 df-inn 8933 df-ndx 12478 df-slot 12479 df-base 12481 df-0g 12724 |
This theorem is referenced by: mndidcl 12850 |
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