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Mirrors > Home > ILE Home > Th. List > 0xp | Unicode version |
Description: The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
0xp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4526 | . . 3 | |
2 | noel 3337 | . . . . . . 7 | |
3 | simprl 505 | . . . . . . 7 | |
4 | 2, 3 | mto 636 | . . . . . 6 |
5 | 4 | nex 1461 | . . . . 5 |
6 | 5 | nex 1461 | . . . 4 |
7 | noel 3337 | . . . 4 | |
8 | 6, 7 | 2false 675 | . . 3 |
9 | 1, 8 | bitri 183 | . 2 |
10 | 9 | eqriv 2114 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1316 wex 1453 wcel 1465 c0 3333 cop 3500 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-in1 588 ax-in2 589 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-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-opab 3960 df-xp 4515 |
This theorem is referenced by: res0 4793 xp0 4928 xpeq0r 4931 xpdisj1 4933 xpima1 4955 xpfi 6786 exmidfodomrlemim 7025 hashxp 10540 |
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