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Mirrors > Home > ILE Home > Th. List > mgmidmo | Unicode version |
Description: A two-sided identity element is unique (if it exists) in any magma. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by NM, 17-Jun-2017.) |
Ref | Expression |
---|---|
mgmidmo |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 |
. . . . 5
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2 | 1 | ralimi 2533 |
. . . 4
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3 | simpr 109 |
. . . . 5
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4 | 3 | ralimi 2533 |
. . . 4
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5 | oveq1 5860 |
. . . . . . . . 9
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6 | id 19 |
. . . . . . . . 9
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7 | 5, 6 | eqeq12d 2185 |
. . . . . . . 8
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8 | 7 | rspcva 2832 |
. . . . . . 7
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9 | oveq2 5861 |
. . . . . . . . 9
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10 | id 19 |
. . . . . . . . 9
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11 | 9, 10 | eqeq12d 2185 |
. . . . . . . 8
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12 | 11 | rspcva 2832 |
. . . . . . 7
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13 | 8, 12 | sylan9req 2224 |
. . . . . 6
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14 | 13 | an42s 584 |
. . . . 5
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15 | 14 | ex 114 |
. . . 4
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16 | 2, 4, 15 | syl2ani 406 |
. . 3
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17 | 16 | rgen2 2556 |
. 2
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18 | oveq1 5860 |
. . . . 5
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19 | 18 | eqeq1d 2179 |
. . . 4
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20 | 19 | ovanraleqv 5877 |
. . 3
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21 | 20 | rmo4 2923 |
. 2
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22 | 17, 21 | mpbir 145 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rmo 2456 df-v 2732 df-un 3125 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 |
This theorem is referenced by: ismgmid 12631 mndideu 12662 |
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