Theorem 0xp 4808

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.)

Assertion

Ref	Expression
0xp	|- ( (/) X. A ) = (/)

Proof of Theorem 0xp

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 4744	. . 3 |- ( z e. ( (/) X. A ) <-> E. x E. y ( z = <. x , y >. /\ ( x e. (/) /\ y e. A ) ) )
2		noel 3497	. . . . . . 7 |- -. x e. (/)
3		simprl 531	. . . . . . 7 |- ( ( z = <. x , y >. /\ ( x e. (/) /\ y e. A ) ) -> x e. (/) )
4	2, 3	mto 668	. . . . . 6 |- -. ( z = <. x , y >. /\ ( x e. (/) /\ y e. A ) )
5	4	nex 1548	. . . . 5 |- -. E. y ( z = <. x , y >. /\ ( x e. (/) /\ y e. A ) )
6	5	nex 1548	. . . 4 |- -. E. x E. y ( z = <. x , y >. /\ ( x e. (/) /\ y e. A ) )
7		noel 3497	. . . 4 |- -. z e. (/)
8	6, 7	2false 708	. . 3 |- ( E. x E. y ( z = <. x , y >. /\ ( x e. (/) /\ y e. A ) ) <-> z e. (/) )
9	1, 8	bitri 184	. 2 |- ( z e. ( (/) X. A ) <-> z e. (/) )
10	9	eqriv 2227	1 |- ( (/) X. A ) = (/)

Syntax hints:   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2201   (/)c0 3493   <.cop 3673   X. cxp 4725

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-opab 4152  df-xp 4733

This theorem is referenced by:  res0  5019  xp0  5158  xpeq0r  5161  xpdisj1  5163  xpima1  5185  xpfi  7129  exmidfodomrlemim  7417  hashxp  11096