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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpm 4960 | . . . . . . . 8 | |
2 | dmxpm 4759 | . . . . . . . . 9 | |
3 | 2 | adantl 275 | . . . . . . . 8 |
4 | 1, 3 | sylbir 134 | . . . . . . 7 |
5 | 4 | adantr 274 | . . . . . 6 |
6 | dmss 4738 | . . . . . . 7 | |
7 | 6 | adantl 275 | . . . . . 6 |
8 | 5, 7 | eqsstrrd 3134 | . . . . 5 |
9 | dmxpss 4969 | . . . . 5 | |
10 | 8, 9 | sstrdi 3109 | . . . 4 |
11 | rnxpm 4968 | . . . . . . . . 9 | |
12 | 11 | adantr 274 | . . . . . . . 8 |
13 | 1, 12 | sylbir 134 | . . . . . . 7 |
14 | 13 | adantr 274 | . . . . . 6 |
15 | rnss 4769 | . . . . . . 7 | |
16 | 15 | adantl 275 | . . . . . 6 |
17 | 14, 16 | eqsstrrd 3134 | . . . . 5 |
18 | rnxpss 4970 | . . . . 5 | |
19 | 17, 18 | sstrdi 3109 | . . . 4 |
20 | 10, 19 | jca 304 | . . 3 |
21 | 20 | ex 114 | . 2 |
22 | xpss12 4646 | . 2 | |
23 | 21, 22 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wss 3071 cxp 4537 cdm 4539 crn 4540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-dm 4549 df-rn 4550 |
This theorem is referenced by: xp11m 4977 |
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