Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4641 | . . 3 | |
2 | an4 560 | . . . . 5 | |
3 | elin 3229 | . . . . . 6 | |
4 | elin 3229 | . . . . . 6 | |
5 | 3, 4 | anbi12i 455 | . . . . 5 |
6 | 2, 5 | bitr4i 186 | . . . 4 |
7 | 6 | opabbii 3965 | . . 3 |
8 | 1, 7 | eqtri 2138 | . 2 |
9 | df-xp 4515 | . . 3 | |
10 | df-xp 4515 | . . 3 | |
11 | 9, 10 | ineq12i 3245 | . 2 |
12 | df-xp 4515 | . 2 | |
13 | 8, 11, 12 | 3eqtr4i 2148 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1316 wcel 1465 cin 3040 copab 3958 cxp 4507 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-opab 3960 df-xp 4515 df-rel 4516 |
This theorem is referenced by: xpindi 4644 xpindir 4645 dmxpin 4731 xpssres 4824 xpdisj1 4933 xpdisj2 4934 imainrect 4954 xpima1 4955 xpima2m 4956 hashxp 10540 txbas 12354 txrest 12372 metreslem 12476 |
Copyright terms: Public domain | W3C validator |