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Mirrors > Home > ILE Home > Th. List > xpeq0r | Unicode version |
Description: A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.) |
Ref | Expression |
---|---|
xpeq0r |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1 4548 | . . 3 | |
2 | 0xp 4614 | . . 3 | |
3 | 1, 2 | syl6eq 2186 | . 2 |
4 | xpeq2 4549 | . . 3 | |
5 | xp0 4953 | . . 3 | |
6 | 4, 5 | syl6eq 2186 | . 2 |
7 | 3, 6 | jaoi 705 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 697 wceq 1331 c0 3358 cxp 4532 |
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 603 ax-in2 604 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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 |
This theorem is referenced by: sqxpeq0 4957 |
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