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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 4615 | . . 3 | |
2 | noel 3408 | . . . . . . 7 | |
3 | simprl 521 | . . . . . . 7 | |
4 | 2, 3 | mto 652 | . . . . . 6 |
5 | 4 | nex 1487 | . . . . 5 |
6 | 5 | nex 1487 | . . . 4 |
7 | noel 3408 | . . . 4 | |
8 | 6, 7 | 2false 691 | . . 3 |
9 | 1, 8 | bitri 183 | . 2 |
10 | 9 | eqriv 2161 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1342 wex 1479 wcel 2135 c0 3404 cop 3573 cxp 4596 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 df-xp 4604 |
This theorem is referenced by: res0 4882 xp0 5017 xpeq0r 5020 xpdisj1 5022 xpima1 5044 xpfi 6886 exmidfodomrlemim 7148 hashxp 10728 |
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