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Mirrors > Home > ILE Home > Th. List > xp0 | 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, 12-Apr-2004.) |
Ref | Expression |
---|---|
xp0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xp 4668 | . . 3 | |
2 | 1 | cnveqi 4763 | . 2 |
3 | cnvxp 5006 | . 2 | |
4 | cnv0 4991 | . 2 | |
5 | 2, 3, 4 | 3eqtr3i 2186 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1335 c0 3395 cxp 4586 ccnv 4587 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4028 df-xp 4594 df-rel 4595 df-cnv 4596 |
This theorem is referenced by: xpeq0r 5010 xpdisj2 5013 djuassen 7154 xpdjuen 7155 0met 12854 |
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