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Mirrors > Home > MPE Home > Th. List > ssrel | Structured 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 3451 |
. . 3
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2 | 1 | alrimivv 1687 |
. 2
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3 | eleq1 2523 |
. . . . . . . . . . 11
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4 | eleq1 2523 |
. . . . . . . . . . 11
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5 | 3, 4 | imbi12d 320 |
. . . . . . . . . 10
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6 | 5 | biimprcd 225 |
. . . . . . . . 9
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7 | 6 | 2alimi 1606 |
. . . . . . . 8
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8 | 19.23vv 1920 |
. . . . . . . 8
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9 | 7, 8 | sylib 196 |
. . . . . . 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 1676 |
. . . 4
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13 | df-rel 4948 |
. . . . 5
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14 | dfss2 3446 |
. . . . 5
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15 | elvv 4998 |
. . . . . . 7
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16 | 15 | imbi2i 312 |
. . . . . 6
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17 | 16 | albii 1611 |
. . . . 5
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18 | 13, 14, 17 | 3bitri 271 |
. . . 4
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19 | dfss2 3446 |
. . . 4
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20 | 12, 18, 19 | 3imtr4g 270 |
. . 3
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21 | 20 | com12 31 |
. 2
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22 | 2, 21 | impbid2 204 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4514 ax-nul 4522 ax-pr 4632 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-v 3073 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-nul 3739 df-if 3893 df-sn 3979 df-pr 3981 df-op 3985 df-opab 4452 df-xp 4947 df-rel 4948 |
This theorem is referenced by: eqrel 5030 relssi 5032 relssdv 5033 cotr 5311 cnvsym 5313 intasym 5314 intirr 5317 codir 5319 qfto 5320 dfso2 27701 dfpo2 27702 dffun10 28082 imagesset 28121 |
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