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Mirrors > Home > ILE Home > Th. List > ssrel2 | Unicode version |
Description: A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4635 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.) |
Ref | Expression |
---|---|
ssrel2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3096 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 1 | a1d 22 |
. . 3
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3 | 2 | ralrimivv 2516 |
. 2
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4 | eleq1 2203 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | eleq1 2203 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 4, 5 | imbi12d 233 |
. . . . . . . . . . 11
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7 | 6 | biimprcd 159 |
. . . . . . . . . 10
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8 | 7 | ralimi 2498 |
. . . . . . . . 9
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9 | 8 | ralimi 2498 |
. . . . . . . 8
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10 | r19.23v 2544 |
. . . . . . . . . 10
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11 | 10 | ralbii 2444 |
. . . . . . . . 9
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12 | r19.23v 2544 |
. . . . . . . . 9
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13 | 11, 12 | bitri 183 |
. . . . . . . 8
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14 | 9, 13 | sylib 121 |
. . . . . . 7
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15 | 14 | com23 78 |
. . . . . 6
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16 | 15 | a2d 26 |
. . . . 5
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17 | 16 | alimdv 1852 |
. . . 4
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18 | dfss2 3091 |
. . . . 5
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19 | elxp2 4565 |
. . . . . . 7
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20 | 19 | imbi2i 225 |
. . . . . 6
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21 | 20 | albii 1447 |
. . . . 5
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22 | 18, 21 | bitri 183 |
. . . 4
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23 | dfss2 3091 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 17, 22, 23 | 3imtr4g 204 |
. . 3
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25 | 24 | com12 30 |
. 2
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26 | 3, 25 | impbid2 142 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-opab 3998 df-xp 4553 |
This theorem is referenced by: (None) |
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