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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4643 |
. . 3
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2 | noel 3426 |
. . . . . . 7
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3 | simprl 529 |
. . . . . . 7
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4 | 2, 3 | mto 662 |
. . . . . 6
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5 | 4 | nex 1500 |
. . . . 5
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6 | 5 | nex 1500 |
. . . 4
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7 | noel 3426 |
. . . 4
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8 | 6, 7 | 2false 701 |
. . 3
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9 | 1, 8 | bitri 184 |
. 2
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10 | 9 | eqriv 2174 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-opab 4065 df-xp 4632 |
This theorem is referenced by: res0 4911 xp0 5048 xpeq0r 5051 xpdisj1 5053 xpima1 5075 xpfi 6928 exmidfodomrlemim 7199 hashxp 10801 |
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