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Mirrors > Home > MPE Home > Th. List > ssrel | Structured version Visualization version 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 3426 |
. . 3
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2 | 1 | alrimivv 1774 |
. 2
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3 | eleq1 2517 |
. . . . . . . . . . 11
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4 | eleq1 2517 |
. . . . . . . . . . 11
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5 | 3, 4 | imbi12d 322 |
. . . . . . . . . 10
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6 | 5 | biimprcd 229 |
. . . . . . . . 9
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7 | 6 | 2alimi 1685 |
. . . . . . . 8
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8 | 19.23vv 1819 |
. . . . . . . 8
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9 | 7, 8 | sylib 200 |
. . . . . . 7
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10 | 9 | com23 81 |
. . . . . 6
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11 | 10 | a2d 29 |
. . . . 5
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12 | 11 | alimdv 1763 |
. . . 4
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13 | df-rel 4841 |
. . . . 5
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14 | dfss2 3421 |
. . . . 5
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15 | elvv 4893 |
. . . . . . 7
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16 | 15 | imbi2i 314 |
. . . . . 6
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17 | 16 | albii 1691 |
. . . . 5
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18 | 13, 14, 17 | 3bitri 275 |
. . . 4
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19 | dfss2 3421 |
. . . 4
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20 | 12, 18, 19 | 3imtr4g 274 |
. . 3
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21 | 20 | com12 32 |
. 2
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22 | 2, 21 | impbid2 208 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pr 4639 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-v 3047 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-sn 3969 df-pr 3971 df-op 3975 df-opab 4462 df-xp 4840 df-rel 4841 |
This theorem is referenced by: eqrel 4924 relssi 4926 relssdv 4927 cotrg 5211 cnvsym 5214 intasym 5215 intirr 5218 codir 5220 qfto 5221 dfso2 30394 dfpo2 30395 dffun10 30681 imagesset 30720 undmrnresiss 36210 cnvssco 36212 |
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