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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 4765 | . . . . . . . 8 | |
2 | dmxpm 4573 | . . . . . . . . 9 | |
3 | 2 | adantl 271 | . . . . . . . 8 |
4 | 1, 3 | sylbir 133 | . . . . . . 7 |
5 | 4 | adantr 270 | . . . . . 6 |
6 | dmss 4552 | . . . . . . 7 | |
7 | 6 | adantl 271 | . . . . . 6 |
8 | 5, 7 | eqsstr3d 3034 | . . . . 5 |
9 | dmxpss 4773 | . . . . 5 | |
10 | 8, 9 | syl6ss 3011 | . . . 4 |
11 | rnxpm 4772 | . . . . . . . . 9 | |
12 | 11 | adantr 270 | . . . . . . . 8 |
13 | 1, 12 | sylbir 133 | . . . . . . 7 |
14 | 13 | adantr 270 | . . . . . 6 |
15 | rnss 4582 | . . . . . . 7 | |
16 | 15 | adantl 271 | . . . . . 6 |
17 | 14, 16 | eqsstr3d 3034 | . . . . 5 |
18 | rnxpss 4774 | . . . . 5 | |
19 | 17, 18 | syl6ss 3011 | . . . 4 |
20 | 10, 19 | jca 300 | . . 3 |
21 | 20 | ex 113 | . 2 |
22 | xpss12 4463 | . 2 | |
23 | 21, 22 | impbid1 140 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 wss 2973 cxp 4361 cdm 4363 crn 4364 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-dm 4373 df-rn 4374 |
This theorem is referenced by: xp11m 4779 |
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