Theorem 0nelelxp 4692

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 4680	. 2 |- ( C e. ( A X. B ) <-> E. x E. y ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) )
2		0nelop 4281	. . . 4 |- -. (/) e. <. x , y >.
3		simpl 109	. . . . 5 |- ( ( C = <. x , y >. /\ ( x e. A /\ y e. B ) ) -> C = <. x , y >. )
4	3	eleq2d 2266	. . . 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 1911	. 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 1506   e. wcel 2167   (/)c0 3450   <.cop 3625   X. cxp 4661

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095  df-xp 4669

This theorem is referenced by:  dmsn0el  5139