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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 4745 | . . . . . . . 8 | |
2 | dmxpm 4555 | . . . . . . . . 9 | |
3 | 2 | adantl 262 | . . . . . . . 8 |
4 | 1, 3 | sylbir 125 | . . . . . . 7 |
5 | 4 | adantr 261 | . . . . . 6 |
6 | dmss 4534 | . . . . . . 7 | |
7 | 6 | adantl 262 | . . . . . 6 |
8 | 5, 7 | eqsstr3d 2980 | . . . . 5 |
9 | dmxpss 4753 | . . . . 5 | |
10 | 8, 9 | syl6ss 2957 | . . . 4 |
11 | rnxpm 4752 | . . . . . . . . 9 | |
12 | 11 | adantr 261 | . . . . . . . 8 |
13 | 1, 12 | sylbir 125 | . . . . . . 7 |
14 | 13 | adantr 261 | . . . . . 6 |
15 | rnss 4564 | . . . . . . 7 | |
16 | 15 | adantl 262 | . . . . . 6 |
17 | 14, 16 | eqsstr3d 2980 | . . . . 5 |
18 | rnxpss 4754 | . . . . 5 | |
19 | 17, 18 | syl6ss 2957 | . . . 4 |
20 | 10, 19 | jca 290 | . . 3 |
21 | 20 | ex 108 | . 2 |
22 | xpss12 4445 | . 2 | |
23 | 21, 22 | impbid1 130 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 wss 2917 cxp 4343 cdm 4345 crn 4346 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-dm 4355 df-rn 4356 |
This theorem is referenced by: xp11m 4759 |
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