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Mirrors > Home > ILE Home > Th. List > ssrel | Unicode version |
Description: A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
ssrel |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3033 |
. . 3
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2 | 1 | alrimivv 1810 |
. 2
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3 | eleq1 2157 |
. . . . . . . . . . 11
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4 | eleq1 2157 |
. . . . . . . . . . 11
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5 | 3, 4 | imbi12d 233 |
. . . . . . . . . 10
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6 | 5 | biimprcd 159 |
. . . . . . . . 9
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7 | 6 | 2alimi 1397 |
. . . . . . . 8
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8 | 19.23vv 1819 |
. . . . . . . 8
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9 | 7, 8 | sylib 121 |
. . . . . . 7
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10 | 9 | com23 78 |
. . . . . 6
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11 | 10 | a2d 26 |
. . . . 5
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12 | 11 | alimdv 1814 |
. . . 4
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13 | df-rel 4474 |
. . . . 5
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14 | dfss2 3028 |
. . . . 5
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15 | elvv 4529 |
. . . . . . 7
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16 | 15 | imbi2i 225 |
. . . . . 6
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17 | 16 | albii 1411 |
. . . . 5
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18 | 13, 14, 17 | 3bitri 205 |
. . . 4
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19 | dfss2 3028 |
. . . 4
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20 | 12, 18, 19 | 3imtr4g 204 |
. . 3
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21 | 20 | com12 30 |
. 2
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22 | 2, 21 | 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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-opab 3922 df-xp 4473 df-rel 4474 |
This theorem is referenced by: eqrel 4556 relssi 4558 relssdv 4559 cotr 4846 cnvsym 4848 intasym 4849 intirr 4851 codir 4853 qfto 4854 |
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