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Mirrors > Home > ILE Home > Th. List > inxp | Unicode version |
Description: The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
inxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inopab 4468 | . . 3 | |
2 | an4 520 | . . . . 5 | |
3 | elin 3126 | . . . . . 6 | |
4 | elin 3126 | . . . . . 6 | |
5 | 3, 4 | anbi12i 433 | . . . . 5 |
6 | 2, 5 | bitr4i 176 | . . . 4 |
7 | 6 | opabbii 3824 | . . 3 |
8 | 1, 7 | eqtri 2060 | . 2 |
9 | df-xp 4351 | . . 3 | |
10 | df-xp 4351 | . . 3 | |
11 | 9, 10 | ineq12i 3136 | . 2 |
12 | df-xp 4351 | . 2 | |
13 | 8, 11, 12 | 3eqtr4i 2070 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wceq 1243 wcel 1393 cin 2916 copab 3817 cxp 4343 |
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-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-opab 3819 df-xp 4351 df-rel 4352 |
This theorem is referenced by: xpindi 4471 xpindir 4472 dmxpinm 4556 xpssres 4645 xpdisj1 4747 xpdisj2 4748 imainrect 4766 xpima1 4767 xpima2m 4768 |
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