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Mirrors > Home > ILE Home > Th. List > ssxpbm | Unicode version |
Description: A cross-product subclass relationship is equivalent to the relationship for its components. (Contributed by Jim Kingdon, 12-Dec-2018.) |
Ref | Expression |
---|---|
ssxpbm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpm 5050 |
. . . . . . . 8
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2 | dmxpm 4847 |
. . . . . . . . 9
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3 | 2 | adantl 277 |
. . . . . . . 8
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4 | 1, 3 | sylbir 135 |
. . . . . . 7
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5 | 4 | adantr 276 |
. . . . . 6
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6 | dmss 4826 |
. . . . . . 7
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7 | 6 | adantl 277 |
. . . . . 6
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8 | 5, 7 | eqsstrrd 3192 |
. . . . 5
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9 | dmxpss 5059 |
. . . . 5
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10 | 8, 9 | sstrdi 3167 |
. . . 4
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11 | rnxpm 5058 |
. . . . . . . . 9
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12 | 11 | adantr 276 |
. . . . . . . 8
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13 | 1, 12 | sylbir 135 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 13 | adantr 276 |
. . . . . 6
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15 | rnss 4857 |
. . . . . . 7
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16 | 15 | adantl 277 |
. . . . . 6
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17 | 14, 16 | eqsstrrd 3192 |
. . . . 5
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18 | rnxpss 5060 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | sstrdi 3167 |
. . . 4
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20 | 10, 19 | jca 306 |
. . 3
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21 | 20 | ex 115 |
. 2
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22 | xpss12 4733 |
. 2
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23 | 21, 22 | impbid1 142 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-xp 4632 df-rel 4633 df-cnv 4634 df-dm 4636 df-rn 4637 |
This theorem is referenced by: xp11m 5067 |
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