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Mirrors > Home > ILE Home > Th. List > ssrelrel | Unicode version |
Description: A subclass relationship determined by ordered triples. Use relrelss 5023 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
ssrelrel |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3057 |
. . . 4
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2 | 1 | alrimiv 1828 |
. . 3
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3 | 2 | alrimivv 1829 |
. 2
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4 | elvvv 4562 |
. . . . . . . 8
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5 | eleq1 2177 |
. . . . . . . . . . . . . 14
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6 | eleq1 2177 |
. . . . . . . . . . . . . 14
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7 | 5, 6 | imbi12d 233 |
. . . . . . . . . . . . 13
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8 | 7 | biimprcd 159 |
. . . . . . . . . . . 12
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9 | 8 | alimi 1414 |
. . . . . . . . . . 11
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10 | 19.23v 1837 |
. . . . . . . . . . 11
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11 | 9, 10 | sylib 121 |
. . . . . . . . . 10
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12 | 11 | 2alimi 1415 |
. . . . . . . . 9
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13 | 19.23vv 1838 |
. . . . . . . . 9
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14 | 12, 13 | sylib 121 |
. . . . . . . 8
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15 | 4, 14 | syl5bi 151 |
. . . . . . 7
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16 | 15 | com23 78 |
. . . . . 6
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17 | 16 | a2d 26 |
. . . . 5
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18 | 17 | alimdv 1833 |
. . . 4
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19 | dfss2 3052 |
. . . 4
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20 | dfss2 3052 |
. . . 4
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21 | 18, 19, 20 | 3imtr4g 204 |
. . 3
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22 | 21 | com12 30 |
. 2
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23 | 3, 22 | impbid2 142 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-opab 3950 df-xp 4505 |
This theorem is referenced by: eqrelrel 4600 |
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