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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 5009 | . . . . . . . 8 | |
2 | dmxpm 4808 | . . . . . . . . 9 | |
3 | 2 | adantl 275 | . . . . . . . 8 |
4 | 1, 3 | sylbir 134 | . . . . . . 7 |
5 | 4 | adantr 274 | . . . . . 6 |
6 | dmss 4787 | . . . . . . 7 | |
7 | 6 | adantl 275 | . . . . . 6 |
8 | 5, 7 | eqsstrrd 3165 | . . . . 5 |
9 | dmxpss 5018 | . . . . 5 | |
10 | 8, 9 | sstrdi 3140 | . . . 4 |
11 | rnxpm 5017 | . . . . . . . . 9 | |
12 | 11 | adantr 274 | . . . . . . . 8 |
13 | 1, 12 | sylbir 134 | . . . . . . 7 |
14 | 13 | adantr 274 | . . . . . 6 |
15 | rnss 4818 | . . . . . . 7 | |
16 | 15 | adantl 275 | . . . . . 6 |
17 | 14, 16 | eqsstrrd 3165 | . . . . 5 |
18 | rnxpss 5019 | . . . . 5 | |
19 | 17, 18 | sstrdi 3140 | . . . 4 |
20 | 10, 19 | jca 304 | . . 3 |
21 | 20 | ex 114 | . 2 |
22 | xpss12 4695 | . 2 | |
23 | 21, 22 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wex 1472 wcel 2128 wss 3102 cxp 4586 cdm 4588 crn 4589 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4028 df-xp 4594 df-rel 4595 df-cnv 4596 df-dm 4598 df-rn 4599 |
This theorem is referenced by: xp11m 5026 |
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