Theorem xp0 4743

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.)

Assertion

Ref	Expression
xp0	|- ( A X. (/) ) = (/)

Proof of Theorem xp0

Step	Hyp	Ref	Expression
1		0xp 4420	. . 3 |- ( (/) X. A ) = (/)
2	1	cnveqi 4510	. 2 |- `' ( (/) X. A ) = `' (/)
3		cnvxp 4742	. 2 |- `' ( (/) X. A ) = ( A X. (/) )
4		cnv0 4727	. 2 |- `' (/) = (/)
5	2, 3, 4	3eqtr3i 2068	1 |- ( A X. (/) ) = (/)

Syntax hints:   = wceq 1243   (/)c0 3224   X. cxp 4343   `'ccnv 4344

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353

This theorem is referenced by:  xpeq0r  4746  xpdisj2  4748