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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 3161 |
. . 3
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2 | 1 | alrimivv 1885 |
. 2
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3 | eleq1 2250 |
. . . . . . . . . . 11
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4 | eleq1 2250 |
. . . . . . . . . . 11
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5 | 3, 4 | imbi12d 234 |
. . . . . . . . . 10
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6 | 5 | biimprcd 160 |
. . . . . . . . 9
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7 | 6 | 2alimi 1466 |
. . . . . . . 8
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8 | 19.23vv 1894 |
. . . . . . . 8
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9 | 7, 8 | sylib 122 |
. . . . . . 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 1889 |
. . . 4
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13 | df-rel 4645 |
. . . . 5
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14 | dfss2 3156 |
. . . . 5
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15 | elvv 4700 |
. . . . . . 7
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16 | 15 | imbi2i 226 |
. . . . . 6
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17 | 16 | albii 1480 |
. . . . 5
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18 | 13, 14, 17 | 3bitri 206 |
. . . 4
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19 | dfss2 3156 |
. . . 4
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20 | 12, 18, 19 | 3imtr4g 205 |
. . 3
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21 | 20 | com12 30 |
. 2
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22 | 2, 21 | impbid2 143 |
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-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-opab 4077 df-xp 4644 df-rel 4645 |
This theorem is referenced by: eqrel 4727 relssi 4729 relssdv 4730 cotr 5022 cnvsym 5024 intasym 5025 intirr 5027 codir 5029 qfto 5030 |
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