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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 4671 | . . 3 | |
2 | an4 575 | . . . . 5 | |
3 | elin 3259 | . . . . . 6 | |
4 | elin 3259 | . . . . . 6 | |
5 | 3, 4 | anbi12i 455 | . . . . 5 |
6 | 2, 5 | bitr4i 186 | . . . 4 |
7 | 6 | opabbii 3995 | . . 3 |
8 | 1, 7 | eqtri 2160 | . 2 |
9 | df-xp 4545 | . . 3 | |
10 | df-xp 4545 | . . 3 | |
11 | 9, 10 | ineq12i 3275 | . 2 |
12 | df-xp 4545 | . 2 | |
13 | 8, 11, 12 | 3eqtr4i 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wcel 1480 cin 3070 copab 3988 cxp 4537 |
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-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-opab 3990 df-xp 4545 df-rel 4546 |
This theorem is referenced by: xpindi 4674 xpindir 4675 dmxpin 4761 xpssres 4854 xpdisj1 4963 xpdisj2 4964 imainrect 4984 xpima1 4985 xpima2m 4986 hashxp 10572 txbas 12427 txrest 12445 metreslem 12549 |
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