Theorem 0nelelxp 4688

Description: A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)

Assertion

Ref	Expression
0nelelxp	|- ( C e. ( A X. B ) -> -. (/) e. C )

Proof of Theorem 0nelelxp

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

Step	Hyp	Ref	Expression
1		elxp 4676	. 2 |- ( C e. ( A X. B ) <-> E. x E. y ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) )
2		0nelop 4277	. . . 4 |- -. (/) e. <. x , y >.
3		simpl 109	. . . . 5 |- ( ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> C = <. x , y >. )
4	3	eleq2d 2263	. . . 4 |- ( ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> ( (/) e. C <-> (/) e. <. x , y >. ) )
5	2, 4	mtbiri 676	. . 3 |- ( ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> -. (/) e. C )
6	5	exlimivv 1908	. 2 |- ( E. x E. y ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> -. (/) e. C )
7	1, 6	sylbi 121	1 |- ( C e. ( A X. B ) -> -. (/) e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2164   (/)c0 3446   <.cop 3621   X. cxp 4657

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-opab 4091  df-xp 4665

This theorem is referenced by:  dmsn0el  5135