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Mirrors > Home > MPE 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 3310 |
. . 3
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2 | 1 | alrimivv 1639 |
. 2
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3 | eleq1 2472 |
. . . . . . . . . . 11
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4 | eleq1 2472 |
. . . . . . . . . . 11
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5 | 3, 4 | imbi12d 312 |
. . . . . . . . . 10
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6 | 5 | biimprcd 217 |
. . . . . . . . 9
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7 | 6 | 2alimi 1566 |
. . . . . . . 8
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8 | 19.23vv 1911 |
. . . . . . . 8
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9 | 7, 8 | sylib 189 |
. . . . . . 7
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10 | 9 | com23 74 |
. . . . . 6
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11 | 10 | a2d 24 |
. . . . 5
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12 | 11 | alimdv 1628 |
. . . 4
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13 | df-rel 4852 |
. . . . 5
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14 | dfss2 3305 |
. . . . 5
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15 | elvv 4903 |
. . . . . . 7
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16 | 15 | imbi2i 304 |
. . . . . 6
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17 | 16 | albii 1572 |
. . . . 5
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18 | 13, 14, 17 | 3bitri 263 |
. . . 4
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19 | dfss2 3305 |
. . . 4
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20 | 12, 18, 19 | 3imtr4g 262 |
. . 3
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21 | 20 | com12 29 |
. 2
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22 | 2, 21 | impbid2 196 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: eqrel 4932 relssi 4934 relssdv 4935 cotr 5213 cnvsym 5215 intasym 5216 intirr 5219 codir 5221 qfto 5222 dfso2 25333 dfpo2 25334 dffun10 25675 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2393 ax-sep 4298 ax-nul 4306 ax-pr 4371 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-clab 2399 df-cleq 2405 df-clel 2408 df-nfc 2537 df-ne 2577 df-v 2926 df-dif 3291 df-un 3293 df-in 3295 df-ss 3302 df-nul 3597 df-if 3708 df-sn 3788 df-pr 3789 df-op 3791 df-opab 4235 df-xp 4851 df-rel 4852 |
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